Percolation threshold

The percolation threshold is a mathematical concept in percolation theory that describes the formation of long-range connectivity in random systems. Below the threshold a giant connected component does not exist; while above it, there exists a giant component of the order of system size. In engineering and coffee making, percolation represents the flow of fluids through porous media, but in the mathematics and physics worlds it generally refers to simplified lattice models of random systems or networks (graphs), and the nature of the connectivity in them. The percolation threshold is the critical value of the occupation probability p, or more generally a critical surface for a group of parameters p₁, p₂, ..., such that infinite connectivity (percolation) first occurs.{{Cite book |last1=Stauffer |first1=Dietrich |title=Introduction to percolation theory |last2=Aharony |first2=Amnon |date=2003 |publisher=Taylor & Francis |isbn=978-0-7484-0253-3 |edition=Rev. 2nd |location=London}}

Percolation models

The most common percolation model is to take a regular lattice, like a square lattice, and make it into a random network by randomly "occupying" sites (vertices) or bonds (edges) with a statistically independent probability p. At a critical threshold p_c, large clusters and long-range connectivity first appear, and this is called the percolation threshold. Depending on the method for obtaining the random network, one distinguishes between the site percolation threshold and the bond percolation threshold. More general systems have several probabilities p₁, p₂, etc., and the transition is characterized by a critical surface or manifold. One can also consider continuum systems, such as overlapping disks and spheres placed randomly, or the negative space (Swiss-cheese models).

To understand the threshold, you can consider a quantity such as the probability that there is a continuous path from one boundary to another along occupied sites or bonds—that is, within a single cluster. For example, one can consider a square system, and ask for the probability P that there is a path from the top boundary to the bottom boundary. As a function of the occupation probability p, one finds a sigmoidal plot that goes from P=0 at p=0 to P=1 at p=1. The larger the square is compared to the lattice spacing, the sharper the transition will be. When the system size goes to infinity, P(p) will be a step function at the threshold value p_c. For finite large systems, P(p_c) is a constant whose value depends upon the shape of the system; for the square system discussed above, P(p_c)={{frac|1|2}} exactly for any lattice by a simple symmetry argument.

There are other signatures of the critical threshold. For example, the size distribution (number of clusters of size s) drops off as a power-law for large s at the threshold, n_s(p_c) ~ s^−τ, where τ is a dimension-dependent percolation critical exponents. For an infinite system, the critical threshold corresponds to the first point (as p increases) where the size of the clusters become infinite.

In the systems described so far, it has been assumed that the occupation of a site or bond is completely random—this is the so-called Bernoulli percolation. For a continuum system, random occupancy corresponds to the points being placed by a Poisson process. Further variations involve correlated percolation, such as percolation clusters related to Ising and Potts models of ferromagnets, in which the bonds are put down by the Fortuin–Kasteleyn method.

{{cite journal

| last1 = Kasteleyn

| first1 = P. W.

| first2 = C. M. | last2 = Fortuin

| title = Phase transitions in lattice systems with random local properties

| journal = Journal of the Physical Society of Japan Supplement

| volume = 26

| year = 1969

| pages = 11–14| bibcode = 1969JPSJS..26...11K

}}

In bootstrap or k-sat percolation, sites and/or bonds are first occupied and then successively culled from a system if a site does not have at least k neighbors. Another important model of percolation, in a different universality class altogether, is directed percolation, where connectivity along a bond depends upon the direction of the flow. Another variation of recent interest is Explosive Percolation, whose thresholds are listed on that page.

Over the last several decades, a tremendous amount of work has gone into finding exact and approximate values of the percolation thresholds for a variety of these systems. Exact thresholds are only known for certain two-dimensional lattices that can be broken up into a self-dual array, such that under a triangle-triangle transformation, the system remains the same. Studies using numerical methods have led to numerous improvements in algorithms and several theoretical discoveries.

Simple duality in two dimensions implies that all fully triangulated lattices (e.g., the triangular, union jack, cross dual, martini dual and asanoha or 3-12 dual, and the Delaunay triangulation) all have site thresholds of {{frac|1|2}}, and self-dual lattices (square, martini-B) have bond thresholds of {{frac|1|2}}.

The notation such as (4,8²) comes from Grünbaum and Shephard,

{{cite book|last1=Grünbaum|first1=Branko|last2=Shephard|first2=G. C.|name-list-style=amp|title=Tilings and Patterns|location=New York|publisher=W. H. Freeman|year=1987|isbn=978-0-7167-1193-3|url-access=registration|url=https://archive.org/details/isbn_0716711931}}

and indicates that around a given vertex, going in the clockwise direction, one encounters first a square and then two octagons. Besides the eleven Archimedean lattices composed of regular polygons with every site equivalent, many other more complicated lattices with sites of different classes have been studied.

Error bars in the last digit or digits are shown by numbers in parentheses. Thus, 0.729724(3) signifies 0.729724 ± 0.000003, and 0.74042195(80) signifies 0.74042195 ± 0.00000080. The error bars variously represent one or two standard deviations in net error (including statistical and expected systematic error), or an empirical confidence interval, depending upon the source.

Percolation on networks

For a random tree-like network (i.e., a connected network with no cycle) without degree-degree correlation, it can be shown that such network can have a giant component, and the percolation threshold (transmission probability) is given by

$p_c = \frac{1}{g_1'(1)} = \frac{\langle k \rangle}{\langle k^2 \rangle - \langle k \rangle}$ .

Where $g_1(z)$ is the generating function corresponding to the excess degree distribution, ${\langle k \rangle}$ is the average degree of the network and ${\langle k^2 \rangle}$ is the second moment of the degree distribution. So, for example, for an ER network, since the degree distribution is a Poisson distribution, where ${\langle k^2 \rangle = \langle k \rangle^2 + \langle k \rangle},$ the threshold is at $p_c = {\langle k \rangle}^{-1}$ .

In networks with low clustering, $0 < C \ll 1$ , the critical point gets scaled by $(1-C)^{-1}$ such that:{{Cite journal|last1=Berchenko|first1=Yakir|last2=Artzy-Randrup|first2=Yael|last3=Teicher|first3=Mina|last4=Stone|first4=Lewi|date=2009-03-30|title=Emergence and Size of the Giant Component in Clustered Random Graphs with a Given Degree Distribution|url=https://link.aps.org/doi/10.1103/PhysRevLett.102.138701|journal=Physical Review Letters|language=en|volume=102|issue=13|pages=138701|doi=10.1103/PhysRevLett.102.138701|pmid=19392410 |bibcode=2009PhRvL.102m8701B |issn=0031-9007}}

$p_c = \frac{1}{1-C}\frac{1}{g_1'(1)}.$

This indicates that for a given degree distribution, the clustering leads to a larger percolation threshold, mainly because for a fixed number of links, the clustering structure reinforces the core of the network with the price of diluting the global connections. For networks with high clustering, strong clustering could induce the core–periphery structure, in which the core and periphery might percolate at different critical points, and the above approximate treatment is not applicable.{{Cite journal|last1=Li|first1=Ming|last2=Liu|first2=Run-Ran|last3=Lü|first3=Linyuan|last4=Hu|first4=Mao-Bin|last5=Xu|first5=Shuqi|last6=Zhang|first6=Yi-Cheng|date=2021-04-25|title=Percolation on complex networks: Theory and application|url=https://www.sciencedirect.com/science/article/pii/S0370157320304269|journal=Physics Reports|language=en|volume=907|pages=1–68|doi=10.1016/j.physrep.2020.12.003|arxiv=2101.11761 |bibcode=2021PhR...907....1L |s2cid=231719831 |issn=0370-1573}}

Percolation in 2D

= Thresholds on Archimedean lattices =

[[File:Archimedean-Lattice.png|thumb|600px|left|This is a picture

{{cite book

| last = Parviainen

| first = Robert

| title = Connectivity Properties of Archimedean and Laves Lattices

| publisher = Uppsala Dissertations in Mathematics

| volume = 34

| year = 2005

| pages = 37

| url = http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4251

| isbn = 978-91-506-1751-1

}} of the 11 Archimedean Lattices or Uniform tilings, in which all polygons are regular and each vertex is surrounded by the same sequence of polygons. The notation "(3⁴, 6)", for example, means that every vertex is surrounded by four triangles and one hexagon. Some common names that have been given to these lattices are listed in the table below.]]

class="wikitable"

! Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

3-12 or super-kagome, (3, 12² )

| 3

| 0.807900764... = (1 − 2 sin ({{pi}}/18))^{{frac|1|2}}

| 0.74042195(80), 0.74042077(2), 0.740420800(2), 0.7404207988509(8), 0.740420798850811610(2),

cross, truncated trihexagonal (4, 6, 12)

| 3

| 0.746, 0.750, 0.747806(4), 0.7478008(2)

| 0.6937314(1), 0.69373383(72), 0.693733124922(2)

square octagon, bathroom tile, 4-8, truncated square

(4, 8²)

| 3

0.729, 0.729724(3), 0.7297232(5)

| 0.6768, 0.67680232(63), 0.6768031269(6), 0.6768031243900113(3),

honeycomb (6³)

| 3

| 0.6962(6), 0.697040230(5), 0.6970402(1), 0.6970413(10), 0.697043(3),

| 0.652703645... = 1-2 sin (π/18), 1+ p³-3p²=0

kagome (3, 6, 3, 6)

| 4

| 0.652703645... = 1 − 2 sin({{pi}}/18)

| 0.5244053(3), 0.52440516(10), 0.52440499(2), 0.524404978(5), 0.52440572..., 0.52440500(1), 0.524404999173(3), 0.524404999167439(4) 0.52440499916744820(1)

ruby, rhombitrihexagonal (3, 4, 6, 4)

| 4

| 0.620, 0.621819(3), 0.62181207(7)

| 0.52483258(53), 0.5248311(1), 0.524831461573(1)

square (4⁴)

| 4

| 0.59274(10), 0.59274605079210(2), 0.59274601(2), 0.59274605095(15), 0.59274621(13), 0.592746050786(3),{{Cite journal |last=Mertens |first=Stephan |date=2022 |title=Exact site-percolation probability on the square lattice |url=https://iopscience.iop.org/article/10.1088/1751-8121/ac4195 |journal=Journal of Physics A: Mathematical and Theoretical |volume=55 |issue=33 |pages=334002 |doi=10.1088/1751-8121/ac4195 |issn=1751-8113|arxiv=2109.12102 |bibcode=2022JPhA...55G4002M }} 0.59274621(33), 0.59274598(4), 0.59274605(3), 0.593(1), 0.591(1), 0.569(13), 0.59274(5)

| {{frac|1|2}}

snub hexagonal, maple leaf (3⁴,6)

| 5

| 0.579 0.579498(3)

| 0.43430621(50), 0.43432764(3), 0.4343283172240(6),

snub square, puzzle (3², 4, 3, 4 )

| 5

| 0.550, 0.550806(3)

| 0.41413743(46), 0.4141378476(7), 0.4141378565917(1),

frieze, elongated triangular(3³, 4²)

| 5

| 0.549, 0.550213(3), 0.5502(8)

| 0.4196(6), 0.41964191(43), 0.41964044(1), 0.41964035886369(2)

triangular (3⁶)

| 6

| {{frac|1|2}}

| 0.347296355... = 2 sin ({{pi}}/18), 1 + p³ − 3p = 0

Note: sometimes "hexagonal" is used in place of honeycomb, although in some contexts a triangular lattice is also called a hexagonal lattice. z = bulk coordination number.

= 2D lattices with extended and complex neighborhoods =

In this section, sq-1,2,3 corresponds to square (NN+2NN+3NN), etc. Equivalent to square-2N+3N+4N, sq(1,2,3). tri = triangular, hc = honeycomb.

class="wikitable"

! Lattice

! z

! Site percolation threshold

! Bond percolation threshold

sq-1, sq-2, sq-3, sq-5

| 4

| 0.5927... (square site)

sq-1,2, sq-2,3, sq-3,5... 3x3 square

| 8

| 0.407... (square matching)

| 0.25036834(6), 0.2503685, 0.25036840(4)

sq-1,3

| 8

| 0.337

| 0.2214995

sq-2,5: 2NN+5NN

| 8

| 0.337

hc-1,2,3: honeycomb-NN+2NN+3NN

| 12

| 0.300, 0.300, 0.302960... = 1-p_c(site, hc)

tri-1,2: triangular-NN+2NN

| 12

| 0.295, 0.289, 0.290258(19)

tri-2,3: triangular-2NN+3NN

| 12

| 0.232020(36), 0.232020(20)

sq-4: square-4NN

| 8

| 0.270...

sq-1,5: square-NN+5NN (r ≤ 2)

| 8

| 0.277

sq-1,2,3: square-NN+2NN+3NN

| 12

| 0.292, 0.290(5) 0.289, 0.288, 0.2891226(14)

| 0.1522203

sq-2,3,5: square-2NN+3NN+5NN

| 12

| 0.288

sq-1,4: square-NN+4NN

| 12

| 0.236

sq-2,4: square-2NN+4NN

| 12

| 0.225

tri-4: triangular-4NN

| 12

| 0.192450(36), 0.1924428(50)

hc-2,4: honeycomb-2NN+4NN

| 12

| 0.2374

tri-1,3: triangular-NN+3NN

| 12

| 0.264539(21)

tri-1,2,3: triangular-NN+2NN+3NN

| 18

| 0.225, 0.215, 0.215459(36) 0.2154657(17)

sq-3,4: 3NN+4NN

| 12

| 0.221

sq-1,2,5: NN+2NN+5NN

| 12

| 0.240

| 0.13805374

sq-1,3,5: NN+3NN+5NN

| 12

| 0.233

sq-4,5: 4NN+5NN

| 12

| 0.199

sq-1,2,4: NN+2NN+4NN

| 16

| 0.219

sq-1,3,4: NN+3NN+4NN

| 16

| 0.208

sq-2,3,4: 2NN+3NN+4NN

| 16

| 0.202

sq-1,4,5: NN+4NN+5NN

| 16

| 0.187

sq-2,4,5: 2NN+4NN+5NN

| 16

| 0.182

sq-3,4,5: 3NN+4NN+5NN

| 16

| 0.179

sq-1,2,3,5 asterisk pattern

| 16

| 0.208

| 0.1032177

tri-4,5: 4NN+5NN

| 18

| 0.140250(36),

sq-1,2,3,4: NN+2NN+3NN+4NN (

r \le \sqrt{5}

)

| 20

| 0.19671(9), 0.196, 0.196724(10), 0.1967293(7)

| 0.0841509

sq-1,2,4,5: NN+2NN+4NN+5NN

| 20

| 0.177

sq-1,3,4,5: NN+3NN+4NN+5NN

| 20

| 0.172

sq-2,3,4,5: 2NN+3NN+4NN+5NN

| 20

| 0.167

sq-1,2,3,5,6 asterisk pattern

| 20

| 0.0783110

sq-1,2,3,4,5: NN+2NN+3NN+4NN+5NN (

r \le \sqrt{8}

, also within a 5 x 5 square)

| 24

| 0.164, 0.164, 0.1647124(6)

tri-1,4,5: NN+4NN+5NN

| 24

| 0.131660(36)

sq-1,...,6: NN+...+6NN (r≤3)

| 28

| 0.142, 0.1432551(9)

| 0.0558493

tri-2,3,4,5: 2NN+3NN+4NN+5NN

| 30

| 0.117460(36) 0.135823(27)

tri-1,2,3,4,5: NN+2NN+3NN+4NN+5NN

| 36

| 0.115, 0.115740(36), 0.1157399(58)

sq-1,...,7: NN+...+7NN (

r \le \sqrt{10}

)

| 36

| 0.113, 0.1153481(9)

| 0.04169608

| sq lat, diamond boundary: dist. ≤ 4

| 40

| 0.105(5)

sq-1,...,8: NN+..+8NN (

r \le \sqrt{13}

)

| 44

|0.095, 0.095765(5), 0.09580(2), 0.0957661(9)

sq-1,...,9: NN+..+9NN (r≤4)

| 48

| 0.086

| 0.02974268

sq-1,...,11: NN+...+11NN (

r \le \sqrt{18}

)

| 60

| 0.02301190(3)

sq-1,...,23 (r ≤ 7)

| 148

| 0.008342595

sq-1,...,32: NN+...+32NN (

r \le \sqrt{72}

)

| 224

| 0.0053050415(33)

sq-1,...,86: NN+...+86NN (r≤15)

| 708

| 0.001557644(4)

sq-1,...,141: NN+...+141NN (

r \le \sqrt{389}

)

| 1224

| 0.000880188(90)

sq-1,...,185: NN+...+185NN (r≤23)

| 1652

| 0.000645458(4)

sq-1,...,317: NN+...+317NN (r≤31)

| 3000

| 0.000349601(3)

sq-1,...,413: NN+...+413NN (

r \le \sqrt{1280}

)

| 4016

| 0.0002594722(11)

sq lat, diamond boundary: dist. ≤ 6

| 84

| 0.049(5)

sq lat, diamond boundary: dist. ≤ 8

| 144

| 0.028(5)

sq lat, diamond boundary: dist. ≤ 10

| 220

| 0.019(5)

2x2 touching lattice squares* (same as sq-1,2,3,4)

| 20

| φ_c = 0.58365(2), p_c = 0.196724(10), 0.19671(9),

3x3 touching lattice squares* (same as sq-1,...,8))

| 44

| φ_c = 0.59586(2), p_c = 0.095765(5), 0.09580(2)

4x4 touching lattice squares*

| 76

| φ_c = 0.60648(1), p_c = 0.0566227(15), 0.05665(3),

5x5 touching lattice squares*

| 116

| φ_c = 0.61467(2), p_c = 0.037428(2), 0.03745(2),

6x6 touching lattice squares*

| 220

| p_c = 0.02663(1),

10x10 touching lattice squares*

| 436

| φ_c = 0.63609(2), p_c = 0.0100576(5)

within 11 x 11 square (r=5)

| 120

| 0.01048079(6)

within 15 x 15 square (r=7)

| 224

| 0.005287692(22)

20x20 touching lattice squares*

|1676

| φ_c = 0.65006(2), p_c = 0.0026215(3)

within 31 x 31 square (r=15)

| 960

| 0.001131082(5)

100x100 touching lattice squares*

|40396

| φ_c = 0.66318(2), p_c = 0.000108815(12)

1000x1000 touching lattice squares*

|4003996

| φ_c = 0.66639(1), p_c = 1.09778(6)E-06

Here NN = nearest neighbor, 2NN = second nearest neighbor (or next nearest neighbor), 3NN = third nearest neighbor (or next-next nearest neighbor), etc. These are also called 2N, 3N, 4N respectively in some papers.

For overlapping or touching squares, $p_c$ (site) given here is the net fraction of sites occupied $\phi_c$ similar to the $\phi_c$ in continuum percolation. The case of a 2×2 square is equivalent to percolation of a square lattice NN+2NN+3NN+4NN or sq-1,2,3,4 with threshold $1-(1-\phi_c)^{1/4} = 0.196724(10)\ldots$ with $\phi_c= 0.58365(2)$ . The 3×3 square corresponds to sq-1,2,3,4,5,6,7,8 with z=44 and $p_c=1-(1-\phi_c)^{1/9} = 0.095765(5)\ldots$ . The value of z for a k x k square is (2k+1)²-5.

= 2D distorted lattices =

Here, one distorts a regular lattice of unit spacing by moving vertices uniformly within the box $(x-\alpha,x+\alpha),(y-\alpha,y+\alpha)$ , and considers percolation when sites are within Euclidean distance $d$ of each other.

class="wikitable"

! Lattice

! $\overline z$

! $\alpha$

! $d$

! Site percolation threshold

! Bond percolation threshold

square

| 0.2

| 1.1

| 0.8025(2)

| 0.2

| 1.2

| 0.6667(5)

| 0.1

| 1.1

| 0.6619(1)

= Overlapping shapes on 2D lattices =

Site threshold is number of overlapping objects per lattice site. k is the length (net area). Overlapping squares are shown in the complex neighborhood section. Here z is the coordination number to k-mers of either orientation, with $z = k^2+10k-2$ for $1 \times k$ sticks.

class="wikitable"
System ! k ! z ! Site coverage φ_c ! Site percolation threshold p_c
1 x 2 dimer, square lattice \| 2 \| 22 \| 0.54691 0.5483(2){{cite journal \| last = Jasna \| first = C. K. \| author2= V. Sasidevan \| title = Effect of shape asymmetry on percolation of aligned and overlapping objects on lattices \| journal = Phys. Rev. E \| date = 2024 \| volume = 109 \| issue = \| page = 064118 \| doi = 10.1103/PhysRevE.109.064118 \| pmid = \| arxiv = 2308.12932 }} \| 0.17956(3) 0.18019(9)
1 x 2 aligned dimer, square lattice \| 2 \| 14(?) \| 0.5715(18) \| 0.3454(13)
1 x 3 trimer, square lattice \| 3 \| 37 \| 0.49898 0.50004(64) \| 0.10880(2) 0.1093(2)
1 x 4 stick, square lattice \| 4 \| 54 \| 0.45761 \| 0.07362(2)
1 x 5 stick, square lattice \| 5 \| 73 \| 0.42241 \| 0.05341(1)
1 x 6 stick, square lattice \| 6 \| 94 \| 0.39219 \| 0.04063(2)

class="wikitable"

System

! k

! z

! Site coverage φ_c

! Site percolation threshold p_c

1 x 2 dimer, square lattice

| 2

| 22

| 0.54691

0.5483(2){{cite journal

| last = Jasna

| first = C. K.

| author2= V. Sasidevan

| title = Effect of shape asymmetry on percolation of aligned and overlapping objects on lattices

| journal = Phys. Rev. E

| date = 2024

| volume = 109

| issue =

| page = 064118

| doi = 10.1103/PhysRevE.109.064118

| pmid =

| arxiv = 2308.12932

}}

| 0.17956(3)

0.18019(9)

1 x 2 aligned dimer, square lattice

| 2

| 14(?)

| 0.5715(18)

| 0.3454(13)

1 x 3 trimer, square lattice

| 3

| 37

| 0.49898

0.50004(64)

| 0.10880(2)

0.1093(2)

1 x 4 stick, square lattice

| 4

| 54

| 0.45761

| 0.07362(2)

1 x 5 stick, square lattice

| 5

| 73

| 0.42241

| 0.05341(1)

1 x 6 stick, square lattice

| 6

| 94

| 0.39219

| 0.04063(2)

The coverage is calculated from $p_c$ by $\phi_c = 1-(1-p_c)^{2 k}$ for $1 \times k$ sticks, because there are $2k$ sites where a stick will cause an overlap with a given site.

For aligned $1 \times k$ sticks: $\phi_c = 1-(1-p_c)^{k}$

= Approximate formulas for thresholds of Archimedean lattices =

class="wikitable"

! Lattice

! z

! Site percolation threshold

! Bond percolation threshold

(3, 12² )

| 3

(4, 6, 12)

| 3

(4, 8²)

| 3

| 0.676835..., 4p³ + 3p⁴ − 6 p⁵ − 2 p⁶ = 1

honeycomb (6³)

| 3

kagome (3, 6, 3, 6)

| 4

| 0.524430..., 3p² + 6p³ − 12 p⁴+ 6 p⁵ − p⁶ = 1

(3, 4, 6, 4)

| 4

square (4⁴)

| 4

| {{frac|1|2}} (exact)

(3⁴,6 )

| 5

| 0.434371..., 12p³ + 36p⁴ − 21p⁵ − 327 p⁶ + 69p⁷ + 2532p⁸ − 6533 p⁹ + 8256 p¹⁰ − 6255p¹¹ + 2951p¹² − 837 p¹³ + 126 p¹⁴ − 7p¹⁵ = 1 {{citation needed|date=December 2018}}

snub square, puzzle (3², 4, 3, 4 )

| 5

(3³, 4²)

| 5

triangular (3⁶)

| 6

| {{frac|1|2}} (exact)

= AB percolation and colored percolation in 2D =

In AB percolation, a $p_\mathrm{site}$ is the proportion of A sites among B sites, and bonds are drawn between sites of opposite species. It is also called antipercolation.

In colored percolation, occupied sites are assigned one of $n$ colors with equal probability, and connection is made along bonds between neighbors of different colors.

class="wikitable"

! Lattice

! z

! $\overline z$

! Site percolation threshold

triangular AB

| 6

| 0.2145, 0.21524(34),{{Cite journal |last=Nakanishi |first=H |date=1987 |title=Critical behaviour of AB percolation in two dimensions |url=https://iopscience.iop.org/article/10.1088/0305-4470/20/17/040 |journal=Journal of Physics A: Mathematical and General |volume=20 |issue=17 |pages=6075–6083 |doi=10.1088/0305-4470/20/17/040 |issn=0305-4470}} 0.21564(3){{Cite journal |last=Debierre |first=J -M |last2=Bradley |first2=R M |date=1992|title=Scaling properties of antipercolation hulls on the triangular lattice |url=https://iopscience.iop.org/article/10.1088/0305-4470/25/2/014 |journal=Journal of Physics A: Mathematical and General |volume=25 |issue=2 |pages=335–343 |doi=10.1088/0305-4470/25/2/014 |issn=0305-4470}}

AB on square-covering lattice

| 6

| $1-\sqrt{1-p_c(site,sq)} = 0.361835$ {{Cite journal |last=Wu |first=Xian-Yuan |last2=Popov |first2=S. Yu. |date=2003 |title=On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph |url=http://link.springer.com/10.1023/A:1021091316925 |journal=Journal of Statistical Physics |volume=110 |issue=1/2 |pages=443–449 |doi=10.1023/A:1021091316925}}

square three-color

| 4

| 0.80745(5){{Cite journal |last=Kundu |first=Sumanta |last2=Manna |first2=S. S. |date=2017-05-15 |title=Colored percolation |url=http://link.aps.org/doi/10.1103/PhysRevE.95.052124 |journal=Physical Review E |language=en |volume=95 |issue=5 |doi=10.1103/PhysRevE.95.052124 |issn=2470-0045|arxiv=1709.00887 }}

square four-color

| 4

| 0.73415(4)

square five-color

| 4

| 0.69864(7)

square six-color

| 4

| 0.67751(5)

triangular two-color

| 6

| 0.72890(4)

triangular three-color

| 6

| 0.63005(4)

triangular four-color

| 6

| 0.59092(3)

triangular five-color

| 6

| 0.56991(5)

triangular six-color

| 6

| 0.55679(5)

= Site-bond percolation in 2D =

Site bond percolation. Here $p_s$ is the site occupation probability and $p_b$ is the bond occupation probability, and connectivity is made only if both the sites and bonds along a path are occupied. The criticality condition becomes a curve $f(p_{s},p_{b})$ = 0, and some specific critical pairs $(p_{s},p_{b})$ are listed below.

Square lattice:

class="wikitable"

! Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

square

| 4

| 0.615185(15)

| 0.95

| 0.667280(15)

| 0.85

|0.732100(15)

| 0.75

|0.75

| 0.726195(15)

| 0.815560(15)

| 0.65

| 0.85

| 0.615810(30)

|0.95

|0.533620(15)

Honeycomb (hexagonal) lattice:

class="wikitable"

! Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

honeycomb

| 3

| 0.7275(5)

| 0.95

| 0. 0.7610(5)

| 0.90

|0.7986(5)

| 0.85

|0.80

| 0.8481(5)

|0.8401(5)

| 0.80

| 0.85

| 0.7890(5)

| 0.90

| 0.7377(5)

|0.95

|0.6926(5)

Kagome lattice:

class="wikitable"

! Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

kagome

| 4

| 0.6711(4), 0.67097(3)

| 0.95

| 0.6914(5), 0.69210(2)

| 0.90

|0.7162(5), 0.71626(3)

| 0.85

|0.7428(5), 0.74339(3)

|0.80

| 0.75

| 0.7894(9)

| 0.7757(8), 0.77556(3)

| 0.75

| 0.80

| 0.7152(7)

| 0.81206(3)

| 0.70

| 0.85

| 0.6556(6)

| 0.85519(3)

| 0.65

| 0.90

| 0.6046(5)

| 0.90546(3)

| 0.60

|0.95

|0.5615(4)

| 0.96604(4)

| 0.55

| 0.9854(3)

| 0.53

* For values on different lattices, see "An investigation of site-bond percolation on many lattices".

Approximate formula for site-bond percolation on a honeycomb lattice

class="wikitable"

! Lattice

! z

! $\overline z$

! Threshold

! Notes

(6³) honeycomb

| 3

| $p_b p_s [1 - (\sqrt{p_{bc}}/(3-p_{bc}))(\sqrt{p_b} - \sqrt{p_{bc}})] = p_{bc}$ , When equal: p_s = p_b = 0.82199

| approximate formula, p_s = site prob., p_b = bond prob., p_bc = 1 − 2 sin ({{pi}}/18), exact at p_s=1, p_b=p_bc.

= Archimedean duals (Laves lattices) =

File:Dual Archimedean.png

Laves lattices are the duals to the Archimedean lattices. Drawings from. See also Uniform tilings.

class="wikitable"

! Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

Cairo pentagonal

D(3²,4,3,4)=({{frac|2|3}})(5³)+({{frac|1|3}})(5⁴)

|3,4

|3 {{frac|1|3}}

|0.6501834(2), 0.650184(5)

|0.585863... = 1 − p_c^bond(3²,4,3,4)

Pentagonal D(3³,4²)=({{frac|1|3}})(5⁴)+({{frac|2|3}})(5³)

|3,4

|3 {{frac|1|3}}

|0.6470471(2), 0.647084(5), 0.6471(6)

|0.580358... = 1 − p_c^bond(3³,4²), 0.5800(6)

D(3⁴,6)=({{frac|1|5}})(4⁶)+({{frac|4|5}})(4³)

|3,6

|3 {{frac|3|5}}

|0.639447

|0.565694... = 1 − p_c^bond(3⁴,6 )

dice, rhombille tiling

D(3,6,3,6) = ({{frac|1|3}})(4⁶) + ({{frac|2|3}})(4³)

|3,6

|0.5851(4), 0.585040(5)

|0.475595... = 1 − p_c^bond(3,6,3,6 )

ruby dual

D(3,4,6,4) = ({{frac|1|6}})(4⁶) + ({{frac|2|6}})(4³) + ({{frac|3|6}})(4⁴)

|3,4,6

|0.582410(5)

| 0.475167... = 1 − p_c^bond(3,4,6,4 )

union jack, tetrakis square tiling

D(4,8²) = ({{frac|1|2}})(3⁴) + ({{frac|1|2}})(3⁸)

|4,8

|{{frac|1|2}}

| 0.323197... = 1 − p_c^bond(4,8² )

bisected hexagon, cross dual

D(4,6,12)= ({{frac|1|6}})(3¹²)+({{frac|2|6}})(3⁶)+({{frac|1|2}})(3⁴)

|4,6,12

|{{frac|1|2}}

|0.306266... = 1 − p_c^bond(4,6,12)

asanoha (hemp leaf)

D(3, 12²)=({{frac|2|3}})(3³)+({{frac|1|3}})(3¹²)

| 3,12

| 6

|{{frac|1|2}}

| 0.259579... = 1 − p_c^bond(3, 12²)

= 2-uniform lattices =

Top 3 lattices: #13 #12 #36

Bottom 3 lattices: #34 #37 #11

File:2uni4m1.png

Top 2 lattices: #35 #30

Bottom 2 lattices: #41 #42

File:2uni4m2.png

Top 4 lattices: #22 #23 #21 #20

Bottom 3 lattices: #16 #17 #15

File:2uni4m3.png

Top 2 lattices: #31 #32

Bottom lattice: #33

File:2uni4m4.png

class="wikitable"

! #

! Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

|({{frac|1|2}})(3,4,3,12) + ({{frac|1|2}})(3, 12²)

| 4,3

| 3.5

| 0.7680(2)

| 0.67493252(36){{citation needed|date=June 2019}}

|({{frac|1|3}})(3,4,6,4) + ({{frac|2|3}})(4,6,12)

| 4,3

| 3{{frac|1|3}}

| 0.7157(2)

| 0.64536587(40){{citation needed|date=June 2019}}

|({{frac|1|7}})(3⁶) + ({{frac|6|7}})(3²,4,12)

| 6,4

| 4 {{frac|2|7}}

| 0.6808(2)

| 0.55778329(40){{citation needed|date=June 2019}}

|({{frac|2|3}})(3²,6²) + ({{frac|1|3}})(3,6,3,6)

| 4,4

| 4

| 0.6499(2)

| 0.53632487(40){{citation needed|date=June 2019}}

|({{frac|1|7}})(3⁶) + ({{frac|6|7}})(3²,6²)

| 6,4

| 4 {{frac|2|7}}

| 0.6329(2)

| 0.51707873(70){{citation needed|date=June 2019}}

|({{frac|4|5}})(3,4²,6) + ({{frac|1|5}})(3,6,3,6)

| 4,4

| 4

| 0.6286(2)

| 0.51891529(35){{citation needed|date=June 2019}}

|({{frac|4|5}})(3,4²,6) + ({{frac|1|5}})(3,6,3,6)*

| 4,4

| 4

| 0.6279(2)

| 0.51769462(35){{citation needed|date=June 2019}}

|({{frac|2|3}})(3,4²,6) + ({{frac|1|3}})(3,4,6,4)

| 4,4

| 4

| 0.6221(2)

| 0.51973831(40){{citation needed|date=June 2019}}

|({{frac|1|2}})(3⁴,6) + ({{frac|1|2}})(3²,6²)

| 5,4

| 4.5

| 0.6171(2)

| 0.48921280(37){{citation needed|date=June 2019}}

|({{frac|1|2}})(3³,4²) + ({{frac|1|2}})(3,4,6,4)

| 5,4

| 4.5

| 0.5885(2)

| 0.47229486(38){{citation needed|date=June 2019}}

|({{frac|1|2}})(3²,4,3,4) + ({{frac|1|2}})(3,4,6,4)

| 5,4

| 4.5

| 0.5883(2)

| 0.46573078(72){{citation needed|date=June 2019}}

|({{frac|1|2}})(3³,4²) + ({{frac|1|2}})(4⁴)

| 5,4

| 4.5

| 0.5720(2)

| 0.45844622(40){{citation needed|date=June 2019}}

|({{frac|2|3}})(3³,4²) + ({{frac|1|3}})(4⁴)

| 5,4

| 4 {{frac|2|3}}

| 0.5648(2)

| 0.44528611(40){{citation needed|date=June 2019}}

|({{frac|1|4}})(3⁶) + ({{frac|3|4}})(3⁴,6)

| 6,5

| 5 {{frac|1|4}}

| 0.5607(2)

| 0.41109890(37){{citation needed|date=June 2019}}

|({{frac|1|2}})(3³,4²) + ({{frac|1|2}})(3²,4,3,4)

| 5,5

| 5

| 0.5505(2)

| 0.41628021(35){{citation needed|date=June 2019}}

|({{frac|1|3}})(3³,4²) + ({{frac|2|3}})(3²,4,3,4)

| 5,5

| 5

| 0.5504(2)

| 0.41549285(36){{citation needed|date=June 2019}}

|({{frac|1|7}})(3⁶) + ({{frac|6|7}})(3²,4,3,4)

| 6,5

| 5 {{frac|1|7}}

| 0.5440(2)

| 0.40379585(40){{citation needed|date=June 2019}}

|({{frac|1|2}})(3⁶) + ({{frac|1|2}})(3⁴,6)

| 6,5

| 5.5

| 0.5407(2)

| 0.38914898(35){{citation needed|date=June 2019}}

|({{frac|1|3}})(3⁶) + ({{frac|2|3}})(3³,4²)

| 6,5

| 5 {{frac|1|3}}

| 0.5342(2)

| 0.39491996(40){{citation needed|date=June 2019}}

|({{frac|1|2}})(3⁶) + ({{frac|1|2}})(3³,4²)

| 6,5

| 5.5

| 0.5258(2)

| 0.38285085(38){{citation needed|date=June 2019}}

= Inhomogeneous 2-uniform lattice =

File:2uniformLattice37.pdf

This figure shows something similar to the 2-uniform lattice #37, except the polygons are not all regular—there is a rectangle in the place of the two squares—and the size of the polygons is changed. This lattice is in the isoradial representation in which each polygon is inscribed in a circle of unit radius. The two squares in the 2-uniform lattice must now be represented as a single rectangle in order to satisfy the isoradial condition. The lattice is shown by black edges, and the dual lattice by red dashed lines. The green circles show the isoradial constraint on both the original and dual lattices. The yellow polygons highlight the three types of polygons on the lattice, and the pink polygons highlight the two types of polygons on the dual lattice. The lattice has vertex types ({{frac|1|2}})(3³,4²) + ({{frac|1|2}})(3,4,6,4), while the dual lattice has vertex types ({{frac|1|15}})(4⁶)+({{frac|6|15}})(4²,5²)+({{frac|2|15}})(5³)+({{frac|6|15}})(5²,4). The critical point is where the longer bonds (on both the lattice and dual lattice) have occupation probability p = 2 sin (π/18) = 0.347296... which is the bond percolation threshold on a triangular lattice, and the shorter bonds have occupation probability 1 − 2 sin(π/18) = 0.652703..., which is the bond percolation on a hexagonal lattice. These results follow from the isoradial condition but also follow from applying the star-triangle transformation to certain stars on the honeycomb lattice. Finally, it can be generalized to having three different probabilities in the three different directions, p₁, p₂ and p₃ for the long bonds, and {{nowrap|1 − p₁}}, {{nowrap|1 − p₂}}, and {{nowrap|1 − p₃}} for the short bonds, where p₁, p₂ and p₃ satisfy the critical surface for the inhomogeneous triangular lattice.

= Thresholds on 2D bow-tie and martini lattices =

To the left, center, and right are: the martini lattice, the martini-A lattice, the martini-B lattice. Below: the martini covering/medial lattice, same as the 2×2, 1×1 subnet for kagome-type lattices (removed).

File:Martini.png

Some other examples of generalized bow-tie lattices (a-d) and the duals of the lattices (e-h):

File:Bow-tie.png

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
martini ({{frac\|3\|4}})(3,9²)+({{frac\|1\|4}})(9³) \| 3 \| 3 \| 0.764826..., 1 + p⁴ − 3p³ = 0 \| 0.707107... = 1/{{radic\|2}}
bow-tie (c) \| 3,4 \|3 {{frac\|1\|7}} \| \| 0.672929..., 1 − 2p³ − 2p⁴ − 2p⁵ − 7p⁶ + 18p⁷ + 11p⁸ − 35p⁹ + 21p¹⁰ − 4p¹¹ = 0
bow-tie (d) \| 3,4 \|3 {{frac\|1\|3}} \| \| 0.625457..., 1 − 2p² − 3p³ + 4p⁴ − p⁵ = 0
martini-A ({{frac\|2\|3}})(3,7²)+({{frac\|1\|3}})(3,7³) \|3,4 \|3 {{frac\|1\|3}} \|1/{{radic\|2}} \|0.625457..., 1 − 2p² − 3p³ + 4p⁴ − p⁵ = 0
bow-tie dual (e) \| 3,4 \|3 {{frac\|2\|3}} \| \| 0.595482..., 1-p_c^bond (bow-tie (a))
bow-tie (b) \| 3,4,6 \|3 {{frac\|2\|3}} \| \| 0.533213..., 1 − p − 2p³ -4p⁴-4p⁵+15⁶+ 13p⁷-36p⁸+19p⁹+ p¹⁰ + p¹¹=0
martini covering/medial ({{frac\|1\|2}})(3³,9) + ({{frac\|1\|2}})(3,9,3,9) \| 4 \| 4 \| \| 0.707107... = 1/{{radic\|2}} \| 0.57086651(33){{citation needed\|date=June 2019}}
martini-B ({{frac\|1\|2}})(3,5,3,5²) + ({{frac\|1\|2}})(3,5²) \| 3, 5 \| 4 \| 0.618034... = 2/(1 + {{radic\|5}}), 1- p² − p = 0 \| {{frac\|1\|2}}
bow-tie dual (f) \| 3,4,8 \|4 {{frac\|2\|5}} \| \| 0.466787..., 1 − p_c^bond (bow-tie (b))
bow-tie (a) ({{frac\|1\|2}})(3²,4,3²,4) + ({{frac\|1\|2}})(3,4,3) \| 4,6 \|5 \|0.5472(2), 0.5479148(7) \| 0.404518..., 1 − p − 6p² + 6p³ − p⁵ = 0
bow-tie dual (h) \| 3,6,8 \|5 \| \| 0.374543..., 1 − p_c^bond(bow-tie (d))
bow-tie dual (g) \| 3,6,10 \|5 {{frac\|1\|2}} \| 0.547... = p_c^site(bow-tie(a)) \| 0.327071..., 1 − p_c^bond(bow-tie (c))
martini dual ({{frac\|1\|2}})(3³) + ({{frac\|1\|2}})(3⁹) \| 3,9 \| 6 \| {{frac\|1\|2}} \| 0.292893... = 1 − 1/{{radic\|2}}

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

martini ({{frac|3|4}})(3,9²)+({{frac|1|4}})(9³)

| 3

| 0.764826..., 1 + p⁴ − 3p³ = 0

| 0.707107... = 1/{{radic|2}}

bow-tie (c)

| 3,4

|3 {{frac|1|7}}

| 0.672929..., 1 − 2p³ − 2p⁴ − 2p⁵ − 7p⁶ + 18p⁷ + 11p⁸ − 35p⁹ + 21p¹⁰ − 4p¹¹ = 0

bow-tie (d)

| 3,4

|3 {{frac|1|3}}

| 0.625457..., 1 − 2p² − 3p³ + 4p⁴ − p⁵ = 0

martini-A ({{frac|2|3}})(3,7²)+({{frac|1|3}})(3,7³)

|3,4

|3 {{frac|1|3}}

|1/{{radic|2}}

|0.625457..., 1 − 2p² − 3p³ + 4p⁴ − p⁵ = 0

bow-tie dual (e)

| 3,4

|3 {{frac|2|3}}

| 0.595482..., 1-p_c^bond (bow-tie (a))

bow-tie (b)

| 3,4,6

|3 {{frac|2|3}}

| 0.533213..., 1 − p − 2p³ -4p⁴-4p⁵+15⁶+ 13p⁷-36p⁸+19p⁹+ p¹⁰ + p¹¹=0

martini covering/medial ({{frac|1|2}})(3³,9) + ({{frac|1|2}})(3,9,3,9)

| 4

| | 0.707107... = 1/{{radic|2}}

| 0.57086651(33){{citation needed|date=June 2019}}

martini-B ({{frac|1|2}})(3,5,3,5²) + ({{frac|1|2}})(3,5²)

| 3, 5

| 4

| 0.618034... = 2/(1 + {{radic|5}}), 1- p² − p = 0

| {{frac|1|2}}

bow-tie dual (f)

| 3,4,8

|4 {{frac|2|5}}

| 0.466787..., 1 − p_c^bond (bow-tie (b))

bow-tie (a) ({{frac|1|2}})(3²,4,3²,4) + ({{frac|1|2}})(3,4,3)

| 4,6

|0.5472(2), 0.5479148(7)

| 0.404518..., 1 − p − 6p² + 6p³ − p⁵ = 0

bow-tie dual (h)

| 3,6,8

| 0.374543..., 1 − p_c^bond(bow-tie (d))

bow-tie dual (g)

| 3,6,10

|5 {{frac|1|2}}

| 0.547... = p_c^site(bow-tie(a))

| 0.327071..., 1 − p_c^bond(bow-tie (c))

martini dual ({{frac|1|2}})(3³) + ({{frac|1|2}})(3⁹)

| 3,9

| 6

| {{frac|1|2}}

| 0.292893... = 1 − 1/{{radic|2}}

= Thresholds on 2D covering, medial, and matching lattices =

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
(4, 6, 12) covering/medial \| 4 \| 4 \| p_c^bond(4, 6, 12) = 0.693731... \| 0.5593140(2), 0.559315(1){{citation needed\|date=June 2019}}
(4, 8²) covering/medial, square kagome \| 4 \| 4 \| p_c^bond(4,8²) = 0.676803... \| 0.544798017(4), 0.54479793(34){{citation needed\|date=June 2019}}
(3⁴, 6) medial \| 4 \| 4 \| \| 0.5247495(5)
(3,4,6,4) medial \| 4 \| 4 \| \| 0.51276
(3², 4, 3, 4) medial \| 4 \| 4 \| \| 0.512682929(8)
(3³, 4²) medial \| 4 \| 4 \| \| 0.5125245984(9)
square covering (non-planar) \| 6 \|6 \| {{frac\|1\|2}} \| 0.3371(1)
square matching lattice (non-planar) \| 8 \|8 \| 1 − p_c^site(square) = 0.407253... \| 0.25036834(6)

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

(4, 6, 12) covering/medial

| 4

| p_c^bond(4, 6, 12) = 0.693731...

| 0.5593140(2), 0.559315(1){{citation needed|date=June 2019}}

(4, 8²) covering/medial, square kagome

| 4

| p_c^bond(4,8²) = 0.676803...

| 0.544798017(4), 0.54479793(34){{citation needed|date=June 2019}}

(3⁴, 6) medial

| 4

| 0.5247495(5)

(3,4,6,4) medial

| 4

| 0.51276

(3², 4, 3, 4) medial

| 4

| 0.512682929(8)

(3³, 4²) medial

| 4

| 0.5125245984(9)

square covering (non-planar)

| 6

| {{frac|1|2}}

| 0.3371(1)

square matching lattice (non-planar)

| 8

| 1 − p_c^site(square) = 0.407253...

| 0.25036834(6)

{{multiple image |total_width=600 |align=left |direction=horizontal

|image1=4,6,12covering.svg

|caption1=(4, 6, 12) covering/medial lattice

|image2=4,82coveringlattice.pdf

|caption2=(4, 8²) covering/medial lattice

|image3=312coveringdual.pdf

|caption3=(3,12²) covering/medial lattice (in light grey), equivalent to the kagome (2 × 2) subnet, and in black, the dual of these lattices.

}}

{{multiple image |total_width=400 |align=left |direction=horizontal

|image1=(3,4,6,4) medial lattice.png

|caption1=(3,4,6,4) covering/medial lattice, equivalent to the 2-uniform lattice #30, but with facing triangles made into a diamond. This pattern appears in Iranian tilework.{{cite web |author=Mahmood Maher al-Naqsh |year=1983 |title=MAH 007 |work=The Design and Execution of Drawings in Iranian Tilework |url=http://patterninislamicart.com/drawings-diagrams-analyses/9/design-execution-drawings-iranian-tilework/mah007 |access-date=November 18, 2019 |archive-date=January 9, 2017 |archive-url=https://web.archive.org/web/20170109190100/http://patterninislamicart.com/drawings-diagrams-analyses/9/design-execution-drawings-iranian-tilework/mah007 |url-status=dead }} such as Western tomb tower, Kharraqan.{{cite web |title=Western tomb tower, Kharraqan |url=http://www.tilingsearch.org/HTML/data181/V1P138.html}}

|image2=(3,4,6,4) medial dual.png

|caption2=(3,4,6,4) medial dual, shown in red, with medial lattice in light gray behind it

}}

= Thresholds on 2D chimera non-planar lattices =

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
K(2,2) \| 4 \| 4 \| 0.51253(14) \| 0.44778(15)
K(3,3) \| 6 \| 6 \| 0.43760(15) \| 0.35502(15)
K(4,4) \| 8 \| 8 \| 0.38675(7) \| 0.29427(12)
K(5,5) \| 10 \| 10 \| 0.35115(13) \| 0.25159(13)
K(6,6) \| 12 \| 12 \| 0.32232(13) \| 0.21942(11)
K(7,7) \| 14 \| 14 \| 0.30052(14) \| 0.19475(9)
K(8,8) \| 16 \| 16 \| 0.28103(11) \| 0.17496(10)

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

K(2,2)

| 4

| 0.51253(14)

| 0.44778(15)

K(3,3)

| 6

| 0.43760(15)

| 0.35502(15)

K(4,4)

| 8

| 0.38675(7)

| 0.29427(12)

K(5,5)

| 10

| 0.35115(13)

| 0.25159(13)

K(6,6)

| 12

| 0.32232(13)

| 0.21942(11)

K(7,7)

| 14

| 0.30052(14)

| 0.19475(9)

K(8,8)

| 16

| 0.28103(11)

| 0.17496(10)

= Thresholds on subnet lattices =

File:Kagomesubnets.png

The 2 x 2, 3 x 3, and 4 x 4 subnet kagome lattices. The 2 × 2 subnet is also known as the "triangular kagome" lattice.{{cite journal

| last = Okubo

| first = S.

| title = Submillimeter wave ESR of triangular-kagome antiferromagnet Cu9X2(cpa)6 (X=Cl, Br)

| journal = Physica B

| volume = 246–247

| issue = 2

| year = 1998

| doi = 10.1016/S0921-4526(97)00985-X

| pages = 553–556|bibcode = 1998PhyB..246..553O }}

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
checkerboard – 2 × 2 subnet \| 4,3 \| \| \| 0.596303(1){{cite journal \| last = Haji Akbari \| first = Amir \|author2=R. M. Ziff \| title = Percolation in networks with voids and bottlenecks \| journal = Physical Review E \| volume = 79 \| issue = 2 \| year = 2009 \| pages = 021118 \| doi = 10.1103/PhysRevE.79.021118\| pmid = 19391717 \|bibcode = 2009PhRvE..79b1118H \|arxiv = 0811.4575 \| s2cid = 2554311 }}
checkerboard – 4 × 4 subnet \| 4,3 \| \| \| 0.633685(9)
checkerboard – 8 × 8 subnet \| 4,3 \| \| \| 0.642318(5)
checkerboard – 16 × 16 subnet \| 4,3 \| \| \| 0.64237(1)
checkerboard – 32 × 32 subnet \| 4,3 \| \| \| 0.64219(2)
checkerboard – $\infty$ subnet \| 4,3 \| \| \| 0.642216(10)
kagome – 2 × 2 subnet = (3, 12²) covering/medial \| 4 \| \| p_c^bond (3, 12²) = 0.74042077... \| 0.600861966960(2), 0.6008624(10), 0.60086193(3)
kagome – 3 × 3 subnet \| 4 \| \| \| 0.6193296(10), 0.61933176(5), 0.61933044(32){{citation needed\|date=June 2019}}
kagome – 4 × 4 subnet \| 4 \| \| \| 0.625365(3), 0.62536424(7)
kagome – $\infty$ subnet \| 4 \| \| \| 0.628961(2)
kagome – (1 × 1):(2 × 2) subnet = martini covering/medial \| 4 \| \| p_c^bond(martini) = 1/{{radic\|2}} = 0.707107... \| 0.57086648(36){{citation needed\|date=June 2019}}
kagome – (1 × 1):(3 × 3) subnet \| 4,3 \| \| 0.728355596425196... \| 0.58609776(37){{citation needed\|date=June 2019}}
kagome – (1 × 1):(4 × 4) subnet \| \| \| 0.738348473943256... \|
kagome – (1 × 1):(5 × 5) subnet \| \| \| 0.743548682503071... \|
kagome – (1 × 1):(6 × 6) subnet \| \| \| 0.746418147634282... \|
kagome – (2 × 2):(3 × 3) subnet \| \| \| \| 0.61091770(30){{citation needed\|date=June 2019}}
triangular – 2 × 2 subnet \| 6,4 \| \| \| 0.471628788
triangular – 3 × 3 subnet \| 6,4 \| \| \| 0.509077793
triangular – 4 × 4 subnet \| 6,4 \| \| \| 0.524364822
triangular – 5 × 5 subnet \| 6,4 \| \| \| 0.5315976(10)
triangular – $\infty$ subnet \| 6,4 \| \| \| 0.53993(1)

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

checkerboard – 2 × 2 subnet

| 4,3

| 0.596303(1){{cite journal

| last = Haji Akbari

| first = Amir

|author2=R. M. Ziff

| title = Percolation in networks with voids and bottlenecks

| journal = Physical Review E

| volume = 79

| issue = 2

| year = 2009

| pages = 021118

| doi = 10.1103/PhysRevE.79.021118| pmid = 19391717

|bibcode = 2009PhRvE..79b1118H |arxiv = 0811.4575 | s2cid = 2554311

}}

checkerboard – 4 × 4 subnet

| 4,3

| 0.633685(9)

checkerboard – 8 × 8 subnet

| 4,3

| 0.642318(5)

checkerboard – 16 × 16 subnet

| 4,3

| 0.64237(1)

checkerboard – 32 × 32 subnet

| 4,3

| 0.64219(2)

checkerboard –

\infty

subnet

| 4,3

| 0.642216(10)

kagome – 2 × 2 subnet = (3, 12²) covering/medial

| 4

| p_c^bond (3, 12²) = 0.74042077...

| 0.600861966960(2), 0.6008624(10), 0.60086193(3)

kagome – 3 × 3 subnet

| 4

| 0.6193296(10), 0.61933176(5), 0.61933044(32){{citation needed|date=June 2019}}

kagome – 4 × 4 subnet

| 4

| 0.625365(3), 0.62536424(7)

kagome –

\infty

subnet

| 4

| 0.628961(2)

kagome – (1 × 1):(2 × 2) subnet = martini covering/medial

| 4

| p_c^bond(martini) = 1/{{radic|2}} = 0.707107...

| 0.57086648(36){{citation needed|date=June 2019}}

kagome – (1 × 1):(3 × 3) subnet

| 4,3

| 0.728355596425196...

| 0.58609776(37){{citation needed|date=June 2019}}

kagome – (1 × 1):(4 × 4) subnet

| 0.738348473943256...

kagome – (1 × 1):(5 × 5) subnet

| 0.743548682503071...

kagome – (1 × 1):(6 × 6) subnet

| 0.746418147634282...

kagome – (2 × 2):(3 × 3) subnet

| 0.61091770(30){{citation needed|date=June 2019}}

triangular – 2 × 2 subnet

| 6,4

| 0.471628788

triangular – 3 × 3 subnet

| 6,4

| 0.509077793

triangular – 4 × 4 subnet

| 6,4

| 0.524364822

triangular – 5 × 5 subnet

| 6,4

| 0.5315976(10)

triangular –

\infty

subnet

| 6,4

| 0.53993(1)

= Thresholds of random sequentially adsorbed objects =

(For more results and comparison to the jamming density, see Random sequential adsorption)

class="wikitable"
system ! z ! Site threshold
dimers on a honeycomb lattice \| 3 \| 0.69, 0.6653 {{cite journal \| last = Lebrecht \| first = W. \| author2 = P. M. Centres\|author3 = A. J. Ramirez-Pastor \| title = Analytical approximation of the site percolation thresholds for monomers and dimers on two-dimensional lattices \| journal = Physica A \| volume = 516 \| year = 2019 \| doi = 10.1016/j.physa.2018.10.023 \| pages = 133–143 \| bibcode = 2019PhyA..516..133L \| s2cid = 125418069 }}
dimers on a triangular lattice \| 6 \| 0.4872(8),{{cite journal \| last = Cornette \| first = V. \| author2= A. J. Ramirez-Pastor \|author3=F. Nieto \| title = Dependence of the percolation threshold on the size of the percolating species \| journal = Physica A \| volume = 327 \| issue = 1 \| year = 2003 \| doi =10.1016/S0378-4371(03)00453-9 \| pages = 71–75 \| bibcode =2003PhyA..327...71C\| hdl = 11336/138178 \| hdl-access = free }} 0.4873,
aligned linear dimers on a triangular lattice \| 6 \| 0.5157(2) {{cite journal \| last = Longone \| first = Pablo \| author2= P.M. Centres \|author3= A. J. Ramirez-Pastor \| title = Percolation of aligned rigid rods on two-dimensional triangular lattices \| journal = Physical Review E \| volume = 100 \| issue = 5 \| pages = 052104 \| arxiv = 1906.03966 \| year = 2019\| doi = 10.1103/PhysRevE.100.052104 \| pmid = 31870027 \| bibcode = 2019PhRvE.100e2104L \| s2cid = 182953009 }}
aligned linear 4-mers on a triangular lattice \| 6 \| 0.5220(2)
aligned linear 8-mers on a triangular lattice \| 6 \| 0.5281(5)
aligned linear 12-mers on a triangular lattice \| 6 \| 0.5298(8)
linear 16-mers on a triangular lattice \| 6 \| aligned 0.5328(7)
linear 32-mers on a triangular lattice \| 6 \| aligned 0.5407(6)
linear 64-mers on a triangular lattice \| 6 \| aligned 0.5455(4)
aligned linear 80-mers on a triangular lattice \| 6 \| 0.5500(6)
aligned linear k $\longrightarrow \infty$ on a triangular lattice \| 6 \| 0.582(9)
dimers and 5% impurities, triangular lattice \| 6 \| 0.4832(7){{cite journal \| last = Budinski-Petkovic \| first = Lj \|author2= I. Loncarevic \|author3=Z. M. Jacsik \|author4=S. B. Vrhovac \| title = Jamming and percolation in random sequential adsorption of extended objects on a triangular lattice with quenched impurities \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2016 \| issue = 5 \| year = 2016 \| doi = 10.1088/1742-5468/2016/05/053101 \| pages = 053101\| bibcode =2016JSMTE..05.3101B\| s2cid = 3913989 }}
parallel dimers on a square lattice \| 4 \| 0.5863
dimers on a square lattice \| 4 \| 0.5617,{{cite journal \| last = Cherkasova \| first = V. A. \|author2=Yu. Yu. Tarasevich \|author3=N. I. Lebovka \|author4=N.V. Vygornitskii \| title = Percolation of the aligned dimers on a square lattice \| journal = Eur. Phys. J. B \| volume = 74 \| issue = 2 \| year = 2010 \| doi = 10.1140/epjb/e2010-00089-2 \| pages = 205–209 \|bibcode = 2010EPJB...74..205C \|arxiv = 0912.0778 \| s2cid = 118485353 }} 0.5618(1),{{cite journal \| last = Leroyer \| first = Y. \|author2= E. Pommiers \| title = Monte Carlo analysis of percolation of line segments on a square lattice \| journal = Physical Review B \| volume = 50 \| issue = 5 \| year = 1994 \| doi = 10.1103/PhysRevB.50.2795 \| pmid = 9976520 \| pages = 2795–2799 \| bibcode = 1994PhRvB..50.2795L \| arxiv = cond-mat/9312066 \| s2cid = 119495907 }} 0.562,{{cite journal \| last = Vanderwalle \| first = N. \|author2= S. Galam \|author3= M. Kramer \| title = A new universality for random sequential deposition of needles \| journal = Eur. Phys. J. B \| volume = 14 \| issue = 3 \| year = 2000 \| pages = 407–410 \| doi = 10.1007/s100510051047\| arxiv = cond-mat/0004271\| bibcode = 2000EPJB...14..407V\| s2cid = 11142384 }} 0.5713
linear 3-mers on a square lattice \| 4 \| 0.528
3-site 120° angle, 5% impurities, triangular lattice \| 6 \| 0.4574(9)
3-site triangles, 5% impurities, triangular lattice \| 6 \| 0.5222(9)
linear trimers and 5% impurities, triangular lattice \| 6 \|0.4603(8)
linear 4-mers on a square lattice \| 4 \| 0.504
linear 5-mers on a square lattice \| 4 \| 0.490
linear 6-mers on a square lattice \| 4 \| 0.479
linear 8-mers on a square lattice \| 4 \| 0.474, 0.4697(1)
linear 10-mers on a square lattice \| 4 \| 0.469
linear 16-mers on a square lattice \| 4 \| 0.4639(1)
linear 32-mers on a square lattice \| 4 \| 0.4747(2)

class="wikitable"

system

! z

! Site threshold

dimers on a honeycomb lattice

| 3

| 0.69, 0.6653 {{cite journal

| last = Lebrecht

| first = W.

| author2 = P. M. Centres|author3 = A. J. Ramirez-Pastor

| title = Analytical approximation of the site percolation thresholds for monomers and dimers on two-dimensional lattices

| journal = Physica A

| volume = 516

| year = 2019

| doi = 10.1016/j.physa.2018.10.023

| pages = 133–143 | bibcode = 2019PhyA..516..133L

| s2cid = 125418069

}}

dimers on a triangular lattice

| 6

| 0.4872(8),{{cite journal

| last = Cornette

| first = V.

| author2= A. J. Ramirez-Pastor |author3=F. Nieto

| title = Dependence of the percolation threshold on the size of the percolating species

| journal = Physica A

| volume = 327

| issue = 1

| year = 2003

| doi =10.1016/S0378-4371(03)00453-9

| pages = 71–75 | bibcode =2003PhyA..327...71C| hdl = 11336/138178

| hdl-access = free

}} 0.4873,

aligned linear dimers on a triangular lattice

| 6

| 0.5157(2) {{cite journal

| last = Longone

| first = Pablo

| author2= P.M. Centres |author3= A. J. Ramirez-Pastor

| title = Percolation of aligned rigid rods on two-dimensional triangular lattices

| journal = Physical Review E

| volume = 100

| issue = 5

| pages = 052104

| arxiv = 1906.03966

| year = 2019| doi = 10.1103/PhysRevE.100.052104

| pmid = 31870027

| bibcode = 2019PhRvE.100e2104L

| s2cid = 182953009

}}

aligned linear 4-mers on a triangular lattice

| 6

| 0.5220(2)

aligned linear 8-mers on a triangular lattice

| 6

| 0.5281(5)

aligned linear 12-mers on a triangular lattice

| 6

| 0.5298(8)

linear 16-mers on a triangular lattice

| 6

| aligned 0.5328(7)

linear 32-mers on a triangular lattice

| 6

| aligned 0.5407(6)

linear 64-mers on a triangular lattice

| 6

| aligned 0.5455(4)

aligned linear 80-mers on a triangular lattice

| 6

| 0.5500(6)

aligned linear k

\longrightarrow \infty

on a triangular lattice

| 6

| 0.582(9)

dimers and 5% impurities, triangular lattice

| 6

| 0.4832(7){{cite journal

| last = Budinski-Petkovic

| first = Lj

|author2= I. Loncarevic |author3=Z. M. Jacsik |author4=S. B. Vrhovac

| title = Jamming and percolation in random sequential adsorption of extended objects on a triangular lattice with quenched impurities

| journal = Journal of Statistical Mechanics: Theory and Experiment

| volume = 2016

| issue = 5

| year = 2016

| doi = 10.1088/1742-5468/2016/05/053101

| pages = 053101| bibcode =2016JSMTE..05.3101B| s2cid = 3913989

}}

parallel dimers on a square lattice

| 4

| 0.5863

dimers on a square lattice

| 4

| 0.5617,{{cite journal

| last = Cherkasova

| first = V. A.

|author2=Yu. Yu. Tarasevich |author3=N. I. Lebovka |author4=N.V. Vygornitskii

| title = Percolation of the aligned dimers on a square lattice

| journal = Eur. Phys. J. B

| volume = 74

| issue = 2

| year = 2010

| doi = 10.1140/epjb/e2010-00089-2

| pages = 205–209 |bibcode = 2010EPJB...74..205C |arxiv = 0912.0778 | s2cid = 118485353

}}

0.5618(1),{{cite journal

| last = Leroyer

| first = Y.

|author2= E. Pommiers

| title = Monte Carlo analysis of percolation of line segments on a square lattice

| journal = Physical Review B

| volume = 50

| issue = 5

| year = 1994

| doi = 10.1103/PhysRevB.50.2795

| pmid = 9976520

| pages = 2795–2799 | bibcode = 1994PhRvB..50.2795L

| arxiv = cond-mat/9312066

| s2cid = 119495907

}}

0.562,{{cite journal

| last = Vanderwalle

| first = N.

|author2= S. Galam

|author3= M. Kramer

| title = A new universality for random sequential deposition of needles

| journal = Eur. Phys. J. B

| volume = 14

| issue = 3

| year = 2000

| pages = 407–410

| doi = 10.1007/s100510051047| arxiv = cond-mat/0004271| bibcode = 2000EPJB...14..407V| s2cid = 11142384

}} 0.5713

linear 3-mers on a square lattice

| 4

| 0.528

3-site 120° angle, 5% impurities, triangular lattice

| 6

| 0.4574(9)

3-site triangles, 5% impurities, triangular lattice

| 6

| 0.5222(9)

linear trimers and 5% impurities, triangular lattice

| 6

|0.4603(8)

linear 4-mers on a square lattice

| 4

| 0.504

linear 5-mers on a square lattice

| 4

| 0.490

linear 6-mers on a square lattice

| 4

| 0.479

linear 8-mers on a square lattice

| 4

| 0.474, 0.4697(1)

linear 10-mers on a square lattice

| 4

| 0.469

linear 16-mers on a square lattice

| 4

| 0.4639(1)

linear 32-mers on a square lattice

| 4

| 0.4747(2)

The threshold gives the fraction of sites occupied by the objects when site percolation first takes place (not at full jamming). For longer k-mers see Ref.{{cite journal

| last =Kondrat

| first = Grzegorz

|author2=Andrzej Pękalski

| title = Percolation and jamming in random sequential adsorption of linear segments on a square lattice

| journal = Physical Review E

| volume = 63

| issue = 5

| year = 2001

| pages = 051108

| doi = 10.1103/PhysRevE.63.051108| pmid = 11414888

| arxiv =cond-mat/0102031| bibcode =2001PhRvE..63e1108K| s2cid = 44490067

}}

= Thresholds of full dimer coverings of two dimensional lattices =

Here, we are dealing with networks that are obtained by covering a lattice with dimers, and then consider bond percolation on the remaining bonds. In discrete mathematics, this problem is known as the 'perfect matching' or the 'dimer covering' problem.

class="wikitable"
system ! z ! Bond threshold
Parallel covering, square lattice \| 6 \| 0.381966...{{cite journal \| last = Haji-Akbari \| first = A. \|author2=Nasim Haji-Akbari \|author3=Robert M. Ziff \| title = Dimer Covering and Percolation Frustration \| journal = Physical Review E \| volume = 92 \| issue = 3 \| year = 2015 \| doi = 10.1103/PhysRevE.92.032134 \| pmid = 26465453 \| pages = 032134 \|arxiv = 1507.04411 \|bibcode = 2015PhRvE..92c2134H \| s2cid = 34100812 }}
Shifted covering, square lattice \| 6 \| 0.347296...
Staggered covering, square lattice \| 6 \| 0.376825(2)
Random covering, square lattice \| 6 \| 0.367713(2)
Parallel covering, triangular lattice \| 10 \| 0.237418...
Staggered covering, triangular lattice \| 10 \| 0.237497(2)
Random covering, triangular lattice \| 10 \| 0.235340(1)

class="wikitable"

system

! z

! Bond threshold

Parallel covering, square lattice

| 6

| 0.381966...{{cite journal

| last = Haji-Akbari

| first = A.

|author2=Nasim Haji-Akbari |author3=Robert M. Ziff

| title = Dimer Covering and Percolation Frustration

| journal = Physical Review E

| volume = 92

| issue = 3

| year = 2015

| doi = 10.1103/PhysRevE.92.032134

| pmid = 26465453

| pages = 032134 |arxiv = 1507.04411 |bibcode = 2015PhRvE..92c2134H | s2cid = 34100812

}}

Shifted covering, square lattice

| 6

| 0.347296...

Staggered covering, square lattice

| 6

| 0.376825(2)

Random covering, square lattice

| 6

| 0.367713(2)

Parallel covering, triangular lattice

| 10

| 0.237418...

Staggered covering, triangular lattice

| 10

| 0.237497(2)

Random covering, triangular lattice

| 10

| 0.235340(1)

= Thresholds of polymers (random walks) on a square lattice =

System is composed of ordinary (non-avoiding) random walks of length l on the square lattice.{{cite journal

| last = Zia

| first = R. K. P.

|author2=W. Yong |author3=B. Schmittmann | author3-link = Beate Schmittmann

| title = Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes

| journal = Journal of Mathematical Chemistry

| volume = 45

| year = 2009

| pages = 58–64

| doi = 10.1007/s10910-008-9367-6 | s2cid = 94092783

}}

class="wikitable"
l (polymer length) ! z ! Bond percolation
1 \| 4 \| 0.5(exact){{cite journal \| last = Wu \| first = Yong \|author2=B. Schmittmann \| author2-link = Beate Schmittmann \|author3=R. K. P. Zia \| title = Two-dimensional polymer networks near percolation \| journal = Journal of Physics A \| volume = 41 \| issue = 2 \| year = 2008 \| pages = 025008 \| doi = 10.1088/1751-8113/41/2/025004 \|bibcode = 2008JPhA...41b5004W \| s2cid = 13053653 }}
2 \| 4 \| 0.47697(4)
4 \| 4 \| 0.44892(6)
8 \| 4 \| 0.41880(4)

class="wikitable"

l (polymer length)

! z

! Bond percolation

| 4

| 0.5(exact){{cite journal

| last = Wu

| first = Yong

|author2=B. Schmittmann | author2-link = Beate Schmittmann |author3=R. K. P. Zia

| title = Two-dimensional polymer networks near percolation

| journal = Journal of Physics A

| volume = 41

| issue = 2

| year = 2008

| pages = 025008

| doi = 10.1088/1751-8113/41/2/025004 |bibcode = 2008JPhA...41b5004W | s2cid = 13053653

}}

| 4

| 0.47697(4)

| 4

| 0.44892(6)

| 4

| 0.41880(4)

= Thresholds of self-avoiding walks of length k added by random sequential adsorption =

class="wikitable" border="1"
k ! z ! Site thresholds ! Bond thresholds
1 \| 4 \| 0.593(2){{cite journal \| last = Cornette \| first = V. \| author2 = A.J. Ramirez-Pastor \|author3=F. Nieto \| title = Percolation of polyatomic species on a square lattice \| journal = European Physical Journal B \| volume = 36 \| issue = 3 \| year = 2003 \| pages = 391–399 \| doi = 10.1140/epjb/e2003-00358-1 \|bibcode = 2003EPJB...36..391C \| s2cid = 119852589 }} \| 0.5009(2)
2 \| 4 \| 0.564(2) \| 0.4859(2)
3 \| 4 \| 0.552(2) \| 0.4732(2)
4 \| 4 \| 0.542(2) \| 0.4630(2)
5 \| 4 \| 0.531(2) \| 0.4565(2)
6 \| 4 \| 0.522(2) \| 0.4497(2)
7 \| 4 \| 0.511(2) \| 0.4423(2)
8 \| 4 \| 0.502(2) \| 0.4348(2)
9 \| 4 \| 0.493(2) \| 0.4291(2)
10 \| 4 \| 0.488(2) \| 0.4232(2)
11 \| 4 \| 0.482(2) \| 0.4159(2)
12 \| 4 \| 0.476(2) \| 0.4114(2)
13 \| 4 \| 0.471(2) \| 0.4061(2)
14 \| 4 \| 0.467(2) \| 0.4011(2)
15 \| 4 \| 0.4011(2) \| 0.3979(2)

class="wikitable" border="1"

! z

! Site thresholds

! Bond thresholds

| 4

| 0.593(2){{cite journal

| last = Cornette

| first = V.

| author2 = A.J. Ramirez-Pastor |author3=F. Nieto

| title = Percolation of polyatomic species on a square lattice

| journal = European Physical Journal B

| volume = 36

| issue = 3

| year = 2003

| pages = 391–399

| doi = 10.1140/epjb/e2003-00358-1 |bibcode = 2003EPJB...36..391C | s2cid = 119852589

}}

| 0.5009(2)

| 4

| 0.564(2)

| 0.4859(2)

| 4

| 0.552(2)

| 0.4732(2)

| 4

| 0.542(2)

| 0.4630(2)

| 4

| 0.531(2)

| 0.4565(2)

| 4

| 0.522(2)

| 0.4497(2)

| 4

| 0.511(2)

| 0.4423(2)

| 4

| 0.502(2)

| 0.4348(2)

| 4

| 0.493(2)

| 0.4291(2)

| 4

| 0.488(2)

| 0.4232(2)

| 4

| 0.482(2)

| 0.4159(2)

| 4

| 0.476(2)

| 0.4114(2)

| 4

| 0.471(2)

| 0.4061(2)

| 4

| 0.467(2)

| 0.4011(2)

| 4

| 0.4011(2)

| 0.3979(2)

= Thresholds on 2D inhomogeneous lattices =

class="wikitable"
Lattice ! z ! Site percolation threshold ! Bond percolation threshold
bow-tie with p = {{frac\|1\|2}} on one non-diagonal bond \| 3 \| \| 0.3819654(5), $(3 - \sqrt{5})/2$

class="wikitable"

Lattice

! z

! Site percolation threshold

! Bond percolation threshold

bow-tie with p = {{frac|1|2}} on one non-diagonal bond

| 3

| 0.3819654(5), $(3 - \sqrt{5})/2$

= Thresholds for 2D continuum models =

class="wikitable"
width=12% \| System ! Φ_c ! η_c ! n_c
Disks of radius r \| 0.67634831(2), 0.6763475(6),{{cite journal \| last = Quintanilla \| first = John A. \|author2=R. M. Ziff \| title = Asymmetry in the percolation thresholds of fully penetrable disks with two different radii \| journal = Physical Review E \| volume = 76 \| issue = 5 \| year = 2007 \| pages = 051115 [6 pages] \| doi = 10.1103/PhysRevE.76.051115\| pmid = 18233631 \|bibcode = 2007PhRvE..76e1115Q }} 0.676339(4),{{cite journal \| last = Quintanilla \| first = J \| author2 = S. Torquato \|author3=R. M. Ziff \| title = Efficient measurement of the percolation threshold for fully penetrable discs \| journal = J. Phys. A: Math. Gen. \| volume = 33 \| issue = 42 \| year = 2000 \| pages = L399–L407 \| doi = 10.1088/0305-4470/33/42/104 \|bibcode = 2000JPhA...33L.399Q \| citeseerx = 10.1.1.6.8207 }} 0.6764(4),{{cite journal \| last = Lorenz \| first = B \| author2 = I. Orgzall \|author3=H.-O. Heuer \| title = Universality and cluster structures in continuum models of percolation with two different radius distributions \| journal = J. Phys. A: Math. Gen. \| volume = 26 \| issue = 18 \| year = 1993 \| pages = 4711–4712 \| doi = 10.1088/0305-4470/26/18/032\| bibcode = 1993JPhA...26.4711L}} 0.6766(5),{{cite journal \| last = Rosso \| first = M \| title = Concentration gradient approach to continuum percolation in two dimensions \| journal = J. Phys. A: Math. Gen. \| volume = 22 \| issue = 4 \| year = 1989 \| pages = L131–L136 \| doi =10.1088/0305-4470/22/4/004\| bibcode =1989JPhA...22L.131R}} 0.676(2),{{cite journal \| last = Gawlinski \| first = Edward T \| author2 = H. Eugene Stanley \| title = Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for non-interacting discs \| journal = J. Phys. A: Math. Gen. \| volume = 14 \| issue = 8 \| year = 1981 \| pages = L291–L299 \| doi =10.1088/0305-4470/14/8/007\| bibcode = 1981JPhA...14L.291G }} 0.679, 0.674{{cite journal \| last = Pike \| first = G. E. \| author2 = C. H. Seager \| title = Percolation and conductivity: A computer study I \| journal = Physical Review B \| volume = 10 \| issue = 4 \| year = 1974 \| pages = 1421–1434 \| doi = 10.1103/PhysRevB.10.1421\| bibcode = 1974PhRvB..10.1421P }} 0.676,{{cite journal \|last1=Lin \|first1=Jianjun \|last2=Chen \|first2=Huisu \|title=Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses \|journal=Powder Technology \|date=2019 \|volume=347 \|pages=17–26 \|doi=10.1016/j.powtec.2019.02.036\|s2cid=104332397 }} 0.680{{Cite journal\|last1=Li\|first1=Mingqi\|last2=Chen\|first2=Huisu\|last3=Lin\|first3=Jianjun\|last4=Zhang\|first4=Rongling\|last5=Liu\|first5=Lin\|date=July 2021\|title=Effects of the pore shape polydispersity on the percolation threshold and diffusivity of porous composites: Theoretical and numerical studies\|url=http://dx.doi.org/10.1016/j.powtec.2021.03.055\|journal=Powder Technology\|volume=386\|pages=382–393\|doi=10.1016/j.powtec.2021.03.055\|s2cid=233675695\|issn=0032-5910}} \| 1.1280867(5),{{cite journal \| last = Koza \| first = Zbigniew \| author2 = Piotr Brzeski \| author3 = Grzegorz Kondrat \| title = Percolation of fully penetrable disks using the three-leg cluster method \| journal = J. Phys. A: Math. Theor. \| volume = (in press) \| issue = 16 \| year = 2023 \| page = 165001 \| doi = 10.1088/1751-8121/acc3d0\| s2cid = 257524315 \| doi-access = free \| bibcode = 2023JPhA...56p5001K }} 1.1276(9), 1.12808737(6), 1.128085(2), 1.128059(12), 1.13,{{citation needed\|date=June 2022 \|reason=Citation is for a source that does not exist. Might be in Domb etal, 1972}} 0.8{{cite journal \| last = Gilbert \| first = E. N. \| title = Random Plane Networks \| journal = J. Soc. Indust. Appl. Math. \| volume = 9 \| issue = 4 \| year = 1961 \| pages = 533–543 \| doi = 10.1137/0109045}} \| 1.43632505(10),{{cite journal \| last = Xu \| first = Wenhui \| author2 = Junfeng Wang \| author3 = Hao Hu \| author4 = Youjin Deng \| title = Critical polynomials in the nonplanar and continuum percolation models \| journal = Physical Review E \| year = 2021 \| volume = 103 \| issue = 2 \| page = 022127 \| doi = 10.1103/PhysRevE.103.022127 \|issn=2470-0045 \| pmid = 33736116 \| arxiv = 2010.02887 \| bibcode = 2021PhRvE.103b2127X \| s2cid = 222140792 }} 1.43632545(8), 1.436322(2), 1.436289(16), 1.436320(4),{{cite journal \| last = Tarasevich \| first = Yuri Yu. \| author2 = Andrei V. Eserkepov \| title = Percolation thresholds for discorectangles: numerical estimation for a range of aspect ratios \| journal = Physical Review E \| arxiv = 1910.05072 \| year = 2020 \| volume = 101 \| issue = 2 \| pages = 022108 \| doi = 10.1103/PhysRevE.101.022108 \| pmid = 32168641 \| bibcode = 2020PhRvE.101b2108T \| s2cid = 204401814 }} 1.436323(3),{{cite journal \| last = Li \| first = Jiantong \|author2= Mikael Östling \| title = Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses \| journal = Physica A \| volume = 462 \| year = 2016 \| pages =940–950 \| doi = 10.1016/j.physa.2016.06.020\| bibcode =2016PhyA..462..940L\| url =http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01}} 1.438(2),{{cite journal \| last = Nguyen \| first = Van Lien \|author2= Enrique Canessa \| title = Finite-size scaling in two-dimensional continuum percolation models \| journal = Modern Physics Letters B \| volume = 13 \| issue = 17 \| year = 1999 \| pages = 577–583 \| doi = 10.1142/S0217984999000737\| arxiv = cond-mat/9909200 \| bibcode = 1999MPLB...13..577N \| s2cid = 18560722 }} 1.216 (48){{cite journal \| last = Roberts \| first = F. D. K. \| title = A Monte Carlo Solution of a Two-Dimensional Unstructured Cluster Problem \| journal = Biometrika \| volume = 54 \| issue = 3/4 \| year = 1967 \| pages = 625–628 \| doi = 10.2307/2335053\| jstor = 2335053 \| pmid = 6064024 }}
Ellipses, ε = 1.5 \| 0.0043 \| 0.00431 \| 2.059081(7)
Ellipses, ε = {{frac\|5\|3}} \| 0.65 \| 1.05 \| 2.28
Ellipses, ε = 2 \| 0.6287945(12), 0.63{{cite journal \| last = Xia \| first = W. \|author2=M. F. Thorpe \| title = Percolation properties of random ellipses \| journal = Physical Review A \| volume = 38 \| issue = 5 \| year = 1988 \| pages = 2650–2656 \| doi = 10.1103/PhysRevA.38.2650\|bibcode = 1988PhRvA..38.2650X \| pmid = 9900674 }} \| 0.991000(3), 0.99 \| 2.523560(8), 2.5
Ellipses, ε = 3 \| 0.56 \| 0.82 \| 3.157339(8), 3.14
Ellipses, ε = 4 \| 0.5 \| 0.69 \| 3.569706(8), 3.5
Ellipses, ε = 5 \| 0.455,{{cite journal \| last = Yi \| first = Y.-B. \|author2=A. M. Sastry \| title = Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution \| journal = Proceedings of the Royal Society A \| volume = 460 \| issue = 5 \| year = 2004 \| pages = 2353–2380 \| doi = 10.1098/rspa.2004.1279\|bibcode = 2004RSPSA.460.2353Y \| s2cid = 2475482 }} 0.455, 0.46 \| 0.607 \| 3.861262(12), 3.86
Ellipses, ε = 6 \| \| \| 4.079365(17)
Ellipses, ε = 7 \| \| \| 4.249132(16)
Ellipses, ε = 8 \| \| \| 4.385302(15)
Ellipses, ε = 9 \| \| \| 4.497000(8)
Ellipses, ε = 10 \| 0.301, 0.303, 0.30 \| 0.358 0.36 \| 4.590416(23) 4.56, 4.5
Ellipses, ε = 15 \| \| \| 4.894752(30)
Ellipses, ε = 20 \| 0.178, 0.17 \| 0.196 \| 5.062313(39), 4.99
Ellipses, ε = 50 \| 0.081 \| 0.084 \| 5.393863(28), 5.38
Ellipses, ε = 100 \| 0.0417 \| 0.0426 \| 5.513464(40), 5.42
Ellipses, ε = 200 \| 0.021 \| 0.0212 \| 5.40
{{nowrap\|1=Ellipses, ε = 1000}} \| 0.0043 \| 0.00431 \| 5.624756(22), 5.5
Superellipses, ε = 1, m = 1.5 \| 0.671 \| \|
Superellipses, ε = 2.5, m = 1.5 \| 0.599 \| \|
Superellipses, ε = 5, m = 1.5 \| 0.469 \| \|
Superellipses, ε = 10, m = 1.5 \|0.322 \| \|
disco-rectangles, ε = 1.5 \| \| \| 1.894
disco-rectangles, ε = 2 \| \| \| 2.245
Aligned squares of side $\ell$ \| 0.66675(2), 0.66674349(3),{{cite journal \| last = Mertens \| first = Stephan \|author2=Cristopher Moore \| title = Continuum percolation thresholds in two dimensions \| journal = Physical Review E \| volume = 86 \| issue = 6 \| year = 2012 \| pages = 061109 \| doi = 10.1103/PhysRevE.86.061109\| pmid = 23367895 \|arxiv = 1209.4936 \|bibcode = 2012PhRvE..86f1109M \| s2cid = 15107275 }} 0.66653(1), 0.6666(4),{{cite journal \| last = Baker \| first = Don R. \|author2=Gerald Paul \|author3=Sameet Sreenivasan \|author4=H. Eugene Stanley \| title = Continuum percolation threshold for interpenetrating squares and cubes \| journal = Physical Review E \| volume = 66 \| issue = 4 \| year = 2002 \| pages = 046136 [5 pages] \| doi = 10.1103/PhysRevE.66.046136\| pmid = 12443288 \|arxiv = cond-mat/0203235 \|bibcode = 2002PhRvE..66d6136B \| s2cid = 9561586 }} 0.668 \| 1.09884280(9), 1.0982(3),{{cite journal \| last = Torquato \| first = S. \|author2=Y. Jiao \| title = Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses \| journal = J. Chem. Phys. \| volume = 137 \| issue = 7 \| year = 2012 \| pages = 074106 \| doi = 10.1063/1.4742750 \| pmid = 22920102 \|arxiv = 1208.3720 \|bibcode = 2012JChPh.137g4106T \| s2cid = 13188197 }} 1.098(1) \| 1.09884280(9), 1.0982(3), 1.098(1)
Randomly oriented squares \| 0.62554075(4), 0.6254(2) 0.625, \| 0.9822723(1), 0.9819(6) 0.982278(14){{cite journal \| last = Li \| first = Jiantong \|author2=Mikael Östling \| title = Percolation thresholds of two-dimensional continuum systems of rectangles \| journal = Physical Review E \| volume = 88 \| issue = 1 \| year = 2013 \| pages = 012101 \| doi = 10.1103/PhysRevE.88.012101\| pmid = 23944408 \|bibcode = 2013PhRvE..88a2101L \| s2cid = 21438506 \| url = https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06 }} \| 0.9822723(1), 0.9819(6) 0.982278(14)
Randomly oriented squares within angle $\pi/4$ \| 0.6255(1) \| 0.98216(15) \|
Rectangles, ε = 1.1 \| 0.624870(7) \| 0.980484(19) \| 1.078532(21)
Rectangles, ε = 2 \| 0.590635(5) \| 0.893147(13) \| 1.786294(26)
Rectangles, ε = 3 \| 0.5405983(34) \| 0.777830(7) \| 2.333491(22)
Rectangles, ε = 4 \| 0.4948145(38) \| 0.682830(8) \| 2.731318(30)
Rectangles, ε = 5 \| 0.4551398(31), 0.451 \| 0.607226(6) \| 3.036130(28)
Rectangles, ε = 10 \| 0.3233507(25), 0.319 \| 0.3906022(37) \| 3.906022(37)
Rectangles, ε = 20 \| 0.2048518(22) \| 0.2292268(27) \| 4.584535(54)
Rectangles, ε = 50 \| 0.09785513(36) \| 0.1029802(4) \| 5.149008(20)
Rectangles, ε = 100 \| 0.0523676(6) \| 0.0537886(6) \| 5.378856(60)
Rectangles, ε = 200 \| 0.02714526(34) \| 0.02752050(35) \| 5.504099(69)
Rectangles, ε = 1000 \| 0.00559424(6) \| 0.00560995(6) \| 5.609947(60)
Sticks (needles) of length $\ell$ \| \| \| 5.63726(2),{{cite journal \| last = Li \| first = Jiantong \|author2=Shi-Li Zhang \| title = Finite-size scaling in stick percolation \| journal = Physical Review E \| volume = 80 \| issue = 4 \| year = 2009 \| pages = 040104(R) \| doi = 10.1103/PhysRevE.80.040104\| pmid = 19905260 \|bibcode = 2009PhRvE..80d0104L }} 5.6372858(6), 5.637263(11), 5.63724(18) {{cite journal \| last = Tarasevich \| first = Yuri Yu. \| author2=Andrei V. Eserkepov \| year = 2018 \| title = Percolation of sticks: Effect of stick alignment and length dispersity \| doi=10.1103/PhysRevE.98.062142 \| volume=98 \| issue = 6 \| pages = 062142 \| journal= Physical Review E \| bibcode = 2018PhRvE..98f2142T \| arxiv = 1811.06681 \| s2cid = 54187951 }}
sticks with log-normal length dist. STD=0.5 \| \| \| 4.756(3)
sticks with correlated angle dist. s=0.5 \| \| \| 6.6076(4)
Power-law disks, x = 2.05 \| 0.993(1) {{cite journal \| last = Sasidevan \| first = V. \| year = 2013 \| title = Continuum percolation of overlapping discs with a distribution of radii having a power-law tail \| arxiv = 1302.0085 \| doi=10.1103/PhysRevE.88.022140 \| pmid = 24032808 \| volume=88 \| issue = 2 \| pages = 022140 \| journal=Physical Review E \| bibcode=2013PhRvE..88b2140S \| s2cid = 24046421 }} \| 4.90(1) \| 0.0380(6)
Power-law disks, x = 2.25 \| 0.8591(5) \| 1.959(5) \| 0.06930(12)
Power-law disks, x = 2.5 \| 0.7836(4) \| 1.5307(17) \| 0.09745(11)
Power-law disks, x = 4 \| 0.69543(6) \| 1.18853(19) \| 0.18916(3)
Power-law disks, x = 5 \| 0.68643(13) \| 1.1597(3) \| 0.22149(8)
Power-law disks, x = 6 \| 0.68241(8) \| 1.1470(1) \| 0.24340(5)
Power-law disks, x = 7 \| 0.6803(8) \| 1.140(6) \| 0.25933(16)
Power-law disks, x = 8 \| 0.67917(9) \| 1.1368(5) \| 0.27140(7)
Power-law disks, x = 9 \| 0.67856(12) \| 1.1349(4) \| 0.28098(9)
Voids around disks of radius r \| 1 − Φ_c(disk) = 0.32355169(2), 0.318(2), 0.3261(6){{cite journal \| last = Jin \| first = Yuliang \|author2=Patrick Charbonneau \| title = Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former \| journal = Physical Review E \| volume = 91 \| issue = 4 \| year = 2014 \| pages =042313 \| arxiv = 1409.0688\|bibcode = 2015PhRvE..91d2313J \|doi = 10.1103/PhysRevE.91.042313 \| pmid = 25974497 \| s2cid = 16117644 }} \| \|

class="wikitable"

width=12% | System

! Φ_c

! η_c

! n_c

Disks of radius r

| 0.67634831(2), 0.6763475(6),{{cite journal

| last = Quintanilla

| first = John A.

|author2=R. M. Ziff

| title = Asymmetry in the percolation thresholds of fully penetrable disks with two different radii

| journal = Physical Review E

| volume = 76

| issue = 5

| year = 2007

| pages = 051115 [6 pages]

| doi = 10.1103/PhysRevE.76.051115| pmid = 18233631

|bibcode = 2007PhRvE..76e1115Q }}

0.676339(4),{{cite journal

| last = Quintanilla

| first = J

| author2 = S. Torquato |author3=R. M. Ziff

| title = Efficient measurement of the percolation threshold for fully penetrable discs

| journal = J. Phys. A: Math. Gen.

| volume = 33

| issue = 42

| year = 2000

| pages = L399–L407

| doi = 10.1088/0305-4470/33/42/104 |bibcode = 2000JPhA...33L.399Q | citeseerx = 10.1.1.6.8207

}} 0.6764(4),{{cite journal

| last = Lorenz

| first = B

| author2 = I. Orgzall |author3=H.-O. Heuer

| title = Universality and cluster structures in continuum models of percolation with two different radius distributions

| journal = J. Phys. A: Math. Gen.

| volume = 26

| issue = 18

| year = 1993

| pages = 4711–4712

| doi = 10.1088/0305-4470/26/18/032| bibcode = 1993JPhA...26.4711L}} 0.6766(5),{{cite journal

| last = Rosso

| first = M

| title = Concentration gradient approach to continuum percolation in two dimensions

| journal = J. Phys. A: Math. Gen.

| volume = 22

| issue = 4

| year = 1989

| pages = L131–L136

| doi =10.1088/0305-4470/22/4/004| bibcode =1989JPhA...22L.131R}} 0.676(2),{{cite journal

| last = Gawlinski

| first = Edward T

| author2 = H. Eugene Stanley

| title = Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for non-interacting discs

| journal = J. Phys. A: Math. Gen.

| volume = 14

| issue = 8

| year = 1981

| pages = L291–L299

| doi =10.1088/0305-4470/14/8/007| bibcode = 1981JPhA...14L.291G

}} 0.679, 0.674{{cite journal

| last = Pike

| first = G. E.

| author2 = C. H. Seager

| title = Percolation and conductivity: A computer study I

| journal = Physical Review B

| volume = 10

| issue = 4

| year = 1974

| pages = 1421–1434

| doi = 10.1103/PhysRevB.10.1421| bibcode = 1974PhRvB..10.1421P

}} 0.676,{{cite journal |last1=Lin |first1=Jianjun |last2=Chen |first2=Huisu |title=Measurement of continuum percolation properties of two-dimensional particulate systems comprising congruent and binary superellipses |journal=Powder Technology |date=2019 |volume=347 |pages=17–26 |doi=10.1016/j.powtec.2019.02.036|s2cid=104332397 }}

0.680{{Cite journal|last1=Li|first1=Mingqi|last2=Chen|first2=Huisu|last3=Lin|first3=Jianjun|last4=Zhang|first4=Rongling|last5=Liu|first5=Lin|date=July 2021|title=Effects of the pore shape polydispersity on the percolation threshold and diffusivity of porous composites: Theoretical and numerical studies|url=http://dx.doi.org/10.1016/j.powtec.2021.03.055|journal=Powder Technology|volume=386|pages=382–393|doi=10.1016/j.powtec.2021.03.055|s2cid=233675695|issn=0032-5910}}

| 1.1280867(5),{{cite journal

| last = Koza

| first = Zbigniew

| author2 = Piotr Brzeski

| author3 = Grzegorz Kondrat

| title = Percolation of fully penetrable disks using the three-leg cluster method

| journal = J. Phys. A: Math. Theor.

| volume = (in press)

| issue = 16

| year = 2023

| page = 165001

| doi = 10.1088/1751-8121/acc3d0| s2cid = 257524315

| doi-access = free

| bibcode = 2023JPhA...56p5001K

}} 1.1276(9), 1.12808737(6), 1.128085(2), 1.128059(12), 1.13,{{citation needed|date=June 2022 |reason=Citation is for a source that does not exist. Might be in Domb etal, 1972}} 0.8{{cite journal

| last = Gilbert

| first = E. N.

| title = Random Plane Networks

| journal = J. Soc. Indust. Appl. Math.

| volume = 9

| issue = 4

| year = 1961

| pages = 533–543

| doi = 10.1137/0109045}}

| 1.43632505(10),{{cite journal

| last = Xu

| first = Wenhui

| author2 = Junfeng Wang

| author3 = Hao Hu

| author4 = Youjin Deng

| title = Critical polynomials in the nonplanar and continuum percolation models

| journal = Physical Review E

| year = 2021

| volume = 103

| issue = 2

| page = 022127

| doi = 10.1103/PhysRevE.103.022127 |issn=2470-0045

| pmid = 33736116

| arxiv = 2010.02887

| bibcode = 2021PhRvE.103b2127X

| s2cid = 222140792

}} 1.43632545(8), 1.436322(2), 1.436289(16), 1.436320(4),{{cite journal

| last = Tarasevich | first = Yuri Yu.

| author2 = Andrei V. Eserkepov

| title = Percolation thresholds for discorectangles: numerical estimation for a range of aspect ratios

| journal = Physical Review E

| arxiv = 1910.05072

| year = 2020

| volume = 101

| issue = 2

| pages = 022108

| doi = 10.1103/PhysRevE.101.022108

| pmid = 32168641

| bibcode = 2020PhRvE.101b2108T

| s2cid = 204401814

}} 1.436323(3),{{cite journal

| last = Li

| first = Jiantong

|author2= Mikael Östling

| title = Precise percolation thresholds of two-dimensional random systems comprising overlapping ellipses

| journal = Physica A

| volume = 462

| year = 2016

| pages =940–950

| doi = 10.1016/j.physa.2016.06.020| bibcode =2016PhyA..462..940L| url =http://kth.diva-portal.org/smash/get/diva2:1034150/FULLTEXT01}}

1.438(2),{{cite journal

| last = Nguyen

| first = Van Lien

|author2= Enrique Canessa

| title = Finite-size scaling in two-dimensional continuum percolation models

| journal = Modern Physics Letters B

| volume = 13

| issue = 17

| year = 1999

| pages = 577–583

| doi = 10.1142/S0217984999000737| arxiv = cond-mat/9909200

| bibcode = 1999MPLB...13..577N

| s2cid = 18560722

}}

1.216 (48){{cite journal

| last = Roberts

| first = F. D. K.

| title = A Monte Carlo Solution of a Two-Dimensional Unstructured Cluster Problem

| journal = Biometrika

| volume = 54

| issue = 3/4

| year = 1967

| pages = 625–628

| doi = 10.2307/2335053| jstor = 2335053

| pmid = 6064024

}}

Ellipses, ε = 1.5

| 0.0043

| 0.00431

| 2.059081(7)

Ellipses, ε = {{frac|5|3}}

| 0.65

| 1.05

| 2.28

Ellipses, ε = 2

| 0.6287945(12), 0.63{{cite journal

| last = Xia

| first = W.

|author2=M. F. Thorpe

| title = Percolation properties of random ellipses

| journal = Physical Review A

| volume = 38

| issue = 5

| year = 1988

| pages = 2650–2656

| doi = 10.1103/PhysRevA.38.2650|bibcode = 1988PhRvA..38.2650X

| pmid = 9900674 }}

| 0.991000(3), 0.99

| 2.523560(8), 2.5

Ellipses, ε = 3

| 0.56

| 0.82

| 3.157339(8), 3.14

Ellipses, ε = 4

| 0.5

| 0.69

| 3.569706(8), 3.5

Ellipses, ε = 5

| 0.455,{{cite journal

| last = Yi

| first = Y.-B.

|author2=A. M. Sastry

| title = Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution

| journal = Proceedings of the Royal Society A

| volume = 460

| issue = 5

| year = 2004

| pages = 2353–2380

| doi = 10.1098/rspa.2004.1279|bibcode = 2004RSPSA.460.2353Y | s2cid = 2475482

}} 0.455, 0.46

| 0.607

| 3.861262(12), 3.86

Ellipses, ε = 6

| 4.079365(17)

Ellipses, ε = 7

| 4.249132(16)

Ellipses, ε = 8

| 4.385302(15)

Ellipses, ε = 9

| 4.497000(8)

Ellipses, ε = 10

| 0.301, 0.303, 0.30

| 0.358 0.36

| 4.590416(23) 4.56, 4.5

Ellipses, ε = 15

| 4.894752(30)

Ellipses, ε = 20

| 0.178, 0.17

| 0.196

| 5.062313(39), 4.99

Ellipses, ε = 50

| 0.081

| 0.084

| 5.393863(28), 5.38

Ellipses, ε = 100

| 0.0417

| 0.0426

| 5.513464(40), 5.42

Ellipses, ε = 200

| 0.021

| 0.0212

| 5.40

| 0.0043

| 0.00431

| 5.624756(22), 5.5

Superellipses, ε = 1, m = 1.5

| 0.671

Superellipses, ε = 2.5, m = 1.5

| 0.599

Superellipses, ε = 5, m = 1.5

| 0.469

Superellipses, ε = 10, m = 1.5

|0.322

disco-rectangles, ε = 1.5

| 1.894

disco-rectangles, ε = 2

| 2.245

Aligned squares of side

\ell

| 0.66675(2), 0.66674349(3),{{cite journal

| last = Mertens

| first = Stephan

|author2=Cristopher Moore

| title = Continuum percolation thresholds in two dimensions

| journal = Physical Review E

| volume = 86

| issue = 6

| year = 2012

| pages = 061109

| doi = 10.1103/PhysRevE.86.061109| pmid = 23367895

|arxiv = 1209.4936 |bibcode = 2012PhRvE..86f1109M | s2cid = 15107275

}}

0.66653(1), 0.6666(4),{{cite journal

| last = Baker

| first = Don R.

|author2=Gerald Paul |author3=Sameet Sreenivasan |author4=H. Eugene Stanley

| title = Continuum percolation threshold for interpenetrating squares and cubes

| journal = Physical Review E

| volume = 66

| issue = 4

| year = 2002

| pages = 046136 [5 pages]

| doi = 10.1103/PhysRevE.66.046136| pmid = 12443288

|arxiv = cond-mat/0203235 |bibcode = 2002PhRvE..66d6136B | s2cid = 9561586

}} 0.668

| 1.09884280(9), 1.0982(3),{{cite journal

| last = Torquato

| first = S.

|author2=Y. Jiao

| title = Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. II. Simulation results and analyses

| journal = J. Chem. Phys.

| volume = 137

| issue = 7

| year = 2012

| pages = 074106

| doi = 10.1063/1.4742750 | pmid = 22920102

|arxiv = 1208.3720 |bibcode = 2012JChPh.137g4106T | s2cid = 13188197

}}

1.098(1)

| 1.09884280(9), 1.0982(3), 1.098(1)

Randomly oriented squares

| 0.62554075(4), 0.6254(2) 0.625,

| 0.9822723(1), 0.9819(6) 0.982278(14){{cite journal

| last = Li

| first = Jiantong

|author2=Mikael Östling

| title = Percolation thresholds of two-dimensional continuum systems of rectangles

| journal = Physical Review E

| volume = 88

| issue = 1

| year = 2013

| pages = 012101

| doi = 10.1103/PhysRevE.88.012101| pmid = 23944408

|bibcode = 2013PhRvE..88a2101L | s2cid = 21438506

| url = https://semanticscholar.org/paper/900c43839323029fb14ace3440cc2cb01fc61e06

}}

| 0.9822723(1), 0.9819(6) 0.982278(14)

Randomly oriented squares within angle

\pi/4

| 0.6255(1)

| 0.98216(15)

Rectangles, ε = 1.1

| 0.624870(7)

| 0.980484(19)

| 1.078532(21)

Rectangles, ε = 2

| 0.590635(5)

| 0.893147(13)

| 1.786294(26)

Rectangles, ε = 3

| 0.5405983(34)

| 0.777830(7)

| 2.333491(22)

Rectangles, ε = 4

| 0.4948145(38)

| 0.682830(8)

| 2.731318(30)

Rectangles, ε = 5

| 0.4551398(31), 0.451

| 0.607226(6)

| 3.036130(28)

Rectangles, ε = 10

| 0.3233507(25), 0.319

| 0.3906022(37)

| 3.906022(37)

Rectangles, ε = 20

| 0.2048518(22)

| 0.2292268(27)

| 4.584535(54)

Rectangles, ε = 50

| 0.09785513(36)

| 0.1029802(4)

| 5.149008(20)

Rectangles, ε = 100

| 0.0523676(6)

| 0.0537886(6)

| 5.378856(60)

Rectangles, ε = 200

| 0.02714526(34)

| 0.02752050(35)

| 5.504099(69)

Rectangles, ε = 1000

| 0.00559424(6)

| 0.00560995(6)

| 5.609947(60)

Sticks (needles) of length

\ell

| 5.63726(2),{{cite journal

| last = Li

| first = Jiantong

|author2=Shi-Li Zhang

| title = Finite-size scaling in stick percolation

| journal = Physical Review E

| volume = 80

| issue = 4

| year = 2009

| pages = 040104(R)

| doi = 10.1103/PhysRevE.80.040104| pmid = 19905260

|bibcode = 2009PhRvE..80d0104L }}

5.6372858(6), 5.637263(11), 5.63724(18)

{{cite journal

| last = Tarasevich | first = Yuri Yu.

| author2=Andrei V. Eserkepov

| year = 2018

| title = Percolation of sticks: Effect of stick alignment and length dispersity

| doi=10.1103/PhysRevE.98.062142

| volume=98

| issue = 6

| pages = 062142

| journal= Physical Review E

| bibcode = 2018PhRvE..98f2142T

| arxiv = 1811.06681

| s2cid = 54187951

}}

sticks with log-normal length dist. STD=0.5

| 4.756(3)

sticks with correlated angle dist. s=0.5

| 6.6076(4)

Power-law disks, x = 2.05

| 0.993(1)

{{cite journal

| last = Sasidevan | first = V.

| year = 2013

| title = Continuum percolation of overlapping discs with a distribution of radii having a power-law tail

| arxiv = 1302.0085

| doi=10.1103/PhysRevE.88.022140

| pmid = 24032808

| volume=88

| issue = 2

| pages = 022140

| journal=Physical Review E

| bibcode=2013PhRvE..88b2140S

| s2cid = 24046421

}}

| 4.90(1)

| 0.0380(6)

Power-law disks, x = 2.25

| 0.8591(5)

| 1.959(5)

| 0.06930(12)

Power-law disks, x = 2.5

| 0.7836(4)

| 1.5307(17)

| 0.09745(11)

Power-law disks, x = 4

| 0.69543(6)

| 1.18853(19)

| 0.18916(3)

Power-law disks, x = 5

| 0.68643(13)

| 1.1597(3)

| 0.22149(8)

Power-law disks, x = 6

| 0.68241(8)

| 1.1470(1)

| 0.24340(5)

Power-law disks, x = 7

| 0.6803(8)

| 1.140(6)

| 0.25933(16)

Power-law disks, x = 8

| 0.67917(9)

| 1.1368(5)

| 0.27140(7)

Power-law disks, x = 9

| 0.67856(12)

| 1.1349(4)

| 0.28098(9)

Voids around disks of radius r

| 1 − Φ_c(disk) = 0.32355169(2), 0.318(2), 0.3261(6){{cite journal

| last = Jin

| first = Yuliang

|author2=Patrick Charbonneau

| title = Mapping the arrest of the random Lorentz gas onto the dynamical transition of a simple glass former

| journal = Physical Review E

| volume = 91

| issue = 4

| year = 2014

| pages =042313

| arxiv = 1409.0688|bibcode = 2015PhRvE..91d2313J |doi = 10.1103/PhysRevE.91.042313 | pmid = 25974497

| s2cid = 16117644

}}

File:2D continuum percolation with disks.jpg

File:2D continuum percolation with ellipses of aspect ratio 2.jpg

For disks, $n_c = 4 r^2 N / L^2$ equals the critical number of disks per unit area, measured in units of the diameter $2r$ , where $N$ is the number of objects and $L$ is the system size

For disks, $\eta_c = \pi r^2 N / L^2 = (\pi/4) n_c$ equals critical total disk area.

$4 \eta_c$ gives the number of disk centers within the circle of influence (radius 2 r).

$r_c = L \sqrt{\frac{\eta_c}{\pi N}} = \frac{L}{2} \sqrt{\frac{n_c}{N}}$ is the critical disk radius.

$\eta_c = \pi a b N / L^2$ for ellipses of semi-major and semi-minor axes of a and b, respectively. Aspect ratio $\epsilon = a / b$ with $a > b$ .

$\eta_c = \ell m N / L^2$ for rectangles of dimensions $\ell$ and $m$ . Aspect ratio $\epsilon = \ell/m$ with $\ell > m$ .

$\eta_c = \pi x N / (4 L^2 (x-2))$ for power-law distributed disks with $\hbox{Prob(radius}\ge R) = R^{-x}$ , $R \ge 1$ .

$\phi_c = 1 - e^{-\eta_c}$ equals critical area fraction.

For disks, Ref. use $\phi_c = 1 - e^{-\pi x / 2}$ where $x$ is the density of disks of radius $1/\sqrt{2}$ .

$n_c = \ell^2 N / L^2$ equals number of objects of maximum length $\ell = 2 a$ per unit area.

For ellipses, $n_c = (4 \epsilon / \pi)\eta_c$

For void percolation, $\phi_c = e^{-\eta_c}$ is the critical void fraction.

For more ellipse values, see

For more rectangle values, see

Both ellipses and rectangles belong to the superellipses, with $|x/a|^{2m}+|y/b|^{2m}=1$ . For more percolation values of superellipses, see.

For the monodisperse particle systems, the percolation thresholds of concave-shaped superdisks are obtained as seen in {{cite journal |last1=Lin |first1=Jianjun |last2=Zhang |first2=Wulong |last3=Chen |first3=Huisu |last4=Zhang |first4=Rongling |last5=Liu |first5=Lin |title=Effect of pore characteristic on the percolation threshold and diffusivity of porous media comprising overlapping concave-shaped pores |journal=International Journal of Heat and Mass Transfer |date=2019 |volume=138 |pages=1333–1345|doi=10.1016/j.ijheatmasstransfer.2019.04.110 |s2cid=164424008 }}

For binary dispersions of disks, see {{cite journal

| last = Meeks

| first = Kelsey

| author2=J. Tencer

| author3=M.L. Pantoya

| title = Percolation of binary disk systems: Modeling and theory

| journal = Physical Review E

| volume = 95

| issue = 1

| year = 2017

| pages = 012118

| doi = 10.1103/PhysRevE.95.012118 | pmid = 28208494

| bibcode = 2017PhRvE..95a2118M

| doi-access = free

}}

{{cite journal

| last = Quintanilla

| first = John A.

| title = Measurement of the percolation threshold for fully penetrable disks of different radii

| journal = Physical Review E

| volume = 63

| issue = 6

| year = 2001

| pages = 061108

| doi = 10.1103/PhysRevE.63.061108 | pmid = 11415069

| bibcode = 2001PhRvE..63f1108Q

}}

= Thresholds on 2D random and quasi-lattices =

File:VoronoiDelaunay.svg of points]]

File:Delaunay triangulation example.png

File:VoronoiCov12.png

File:RNGonDelaunayTriangulation128vertices.jpg

File:Gabriel Graph.png

File:UniformInfinitePlanarTriangulation.png

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
Relative neighborhood graph \| \| 2.5576 \| 0.796(2){{cite journal \| last = Melchert \| first = Oliver \| title = Percolation thresholds on planar Euclidean relative-neighborhood graphs \| journal = Physical Review E \| volume = 87 \| issue = 4 \| pages = 042106 \| doi = 10.1103/PhysRevE.87.042106 \| pmid = 23679372 \| year = 2013\|bibcode = 2013PhRvE..87d2106M\|arxiv = 1301.6967\| s2cid = 9691279 }} \| 0.771(2)
Voronoi tessellation \|3 \| \| 0.71410(2),{{cite journal \| last = Becker \| first = A. \|author2=R. M. Ziff \| title = Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations \| journal = Physical Review E \| volume = 80 \| issue = 4 \| pages = 041101 \| doi = 10.1103/PhysRevE.80.041101 \| pmid = 19905267 \| year = 2009\|bibcode = 2009PhRvE..80d1101B \|arxiv = 0906.4360 \| s2cid = 22549508 }} 0.7151* \| 0.68,{{cite journal \| last = Shante \| first = K. S. \|author2=S. Kirkpatrick \| title = An introduction to percolation theory \| journal = Advances in Physics \| volume = 20 \| issue = 85 \| pages = 325–357 \| doi = 10.1080/00018737100101261 \| year = 1971\|bibcode = 1971AdPhy..20..325S }} 0.6670(1), 0.6680(5), 0.666931(5)
Voronoi covering/medial \| 4 \| \| 0.666931(2) \| 0.53618(2)
Randomized kagome/square-octagon, fraction r={{frac\|1\|2}} \| 4 \| \| 0.6599 \|
Penrose rhomb dual \| 4 \| \| 0.6381(3) \| 0.5233(2)
Gabriel graph \| \| 4 \| 0.6348(8), {{cite journal \| last = Norrenbrock \| first = C. \| title = Percolation threshold on planar Euclidean Gabriel Graphs \| journal = Journal of Physics A \| volume = 40 \| issue = 31\| pages = 9253–9258 \| year = 2014 \| arxiv = 0704.2098 \| bibcode = 2007JPhA...40.9253P \| doi = 10.1088/1751-8113/40/31/005 \| s2cid = 680787 }} 0.62{{cite journal \| last = Bertin \| first = E \| author2 = J.-M. Billiot \|author3=R. Drouilhet \| title = Continuum percolation in the Gabriel graph \| journal = Adv. Appl. Probab. \| volume = 34 \| issue = 4 \| pages = 689 \| year = 2002 \| doi=10.1239/aap/1037990948\| s2cid = 121288601 }} \| 0.5167(6), 0.52
Random-line tessellation, dual \| \| 4 \| 0.586(2){{cite journal \| last = Lepage \| first = Thibaut \| author2=Lucie Delaby \| author3=Fausto Malvagi \| author4=Alain Mazzolo \| title =Monte Carlo simulation of fully Markovian stochastic geometries \| journal = Progress in Nuclear Science and Technology \| volume = 2 \| pages = 743–748 \| year = 2011 \| doi=10.15669/pnst.2.743\| doi-access = free }} \|
Penrose rhomb \| \| 4 \| 0.5837(3), 0.0.5610(6) (weighted bonds){{cite journal \| last = Zhang \| first = C. \|author2= K. De'Bell \| title = Reformulation of the percolation problem on a quasilattice: Estimates of the percolation threshold, chemical dimension, and amplitude ratio \| journal = Physical Review B \| volume = 47 \| issue = 14 \| pages = 8558–8564 \| doi = 10.1103/PhysRevB.47.8558 \| year = 1993\| pmid = 10004894 \| bibcode = 1993PhRvB..47.8558Z }} 0.58391(1){{cite journal \| last = Ziff \| first = R. M. \|author2=F. Babalievski \| title = Site percolation on the Penrose rhomb lattice \| journal = Physica A \| volume = 269 \| issue = 2–4 \| pages = 201–210 \| doi = 10.1016/S0378-4371(99)00166-1 \| year = 1999\|bibcode = 1999PhyA..269..201Z }} \| 0.483(5),{{cite journal \| last = Lu \| first = Jian Ping \|author2=Joseph L. Birman \| title = Percolation and Scaling on a Quasilattice \| journal = Journal of Statistical Physics \| volume = 46 \| issue = 5/6 \| pages = 1057–1066 \| doi = 10.1007/BF01011156 \| year = 1987\| bibcode = 1987JSP....46.1057L \| s2cid = 121645524 }} 0.4770(2)
Octagonal lattice, "chemical" links (Ammann–Beenker tiling) \| \|4 \|0.585{{cite journal \| last = Babalievski \| first = F. \| title = Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices \| journal = Physica A \| volume = 220 \| issue = 1995 \| pages = 245–250 \| doi = 10.1016/0378-4371(95)00260-E \| year = 1995 \| bibcode = 1995PhyA..220..245B }} \|0.48
Octagonal lattice, "ferromagnetic" links \| \|5.17 \|0.543 \|0.40
Dodecagonal lattice, "chemical" links \| \|3.63 \|0.628 \|0.54
Dodecagonal lattice, "ferromagnetic" links \| \|4.27 \|0.617 \|0.495
Delaunay triangulation \| \|6 \| {{frac\|1\|2}}{{cite journal \| last = Bollobás \| first = Béla \|author2=Oliver Riordan \| title = The critical probability for random Voronoi percolation in the plane is 1/2 \| journal = Probab. Theory Relat. Fields \| volume = 136 \| issue = 3 \| pages = 417–468 \| doi = 10.1007/s00440-005-0490-z \| year = 2006\| arxiv =math/0410336 \| s2cid = 15985691 }} \| 0.3333(1){{cite journal \| last = Hsu \| first = H. P. \|author2=M. C. Huang \| title = Percolation thresholds, critical exponents, and scaling functions on planar random lattices and their duals \| journal = Physical Review E \| volume = 60 \| issue =6 \| pages = 6361–6370 \| doi = 10.1103/PhysRevE.60.6361 \| pmid = 11970550 \| year = 1999\|bibcode = 1999PhRvE..60.6361H \| s2cid = 8750738 }} 0.3326(5),{{cite journal \| last = Huang \| first = Ming-Chang \|author2= Hsiao-Ping Hsu \| title = Percolation thresholds, critical exponents, and scaling functions on spherical random lattices \| journal = International Journal of Modern Physics C \| volume = 13 \| issue = 3 \| pages = 383–395 \| doi = 10.1142/S012918310200319X \| year = 2002}} 0.333069(2)
Uniform Infinite Planar Triangulation{{cite journal \| last = Angel \| first = Omer \| author2 = Schramm, Oded \| title = Uniform infinite planar triangulation \| journal = Commun. Math. Phys. \| volume = 241 \| issue = 2–3 \| pages = 191–213 \| year = 2003 \| doi = 10.1007/s00220-003-0932-3 \| arxiv = math/0207153\| bibcode = 2003CMaPh.241..191A\| s2cid = 17718301 }} \| \|6 \| {{frac\|1\|2}} \| (2{{radic\|3}} – 1)/11 ≈ 0.2240{{cite journal \| last = Bernardi \| first = Olivier \| author2 = Curien, Nicolas \| author3 = Miermont, Grėgory \| title = A Boltzmann approach to percolation on random triangulations \| journal = Canadian Journal of Mathematics \| volume = 71 \| pages = 1–43 \| year = 2019 \| arxiv = 1705.04064\| doi = 10.4153/CJM-2018-009-x \| s2cid = 6817693 }} {{cite journal \| last = Angel \| first = O. \| author2 = Curien, Nicolas \| title = Percolations on random maps I: Half-plane models \| journal = Annales de l'Institut Henri Poincaré, Probabilités et Statistiques \| volume = 51 \| issue = 2 \| pages = 405–431 \| year = 2014 \| doi = 10.1214/13-AIHP583 \| arxiv = 1301.5311 \| bibcode = 2015AIHPB..51..405A\| s2cid = 14964345 }}

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

Relative neighborhood graph

| 2.5576

| 0.796(2){{cite journal

| last = Melchert

| first = Oliver

| title = Percolation thresholds on planar Euclidean relative-neighborhood graphs

| journal = Physical Review E

| volume = 87

| issue = 4

| pages = 042106

| doi = 10.1103/PhysRevE.87.042106

| pmid = 23679372

| year = 2013|bibcode = 2013PhRvE..87d2106M|arxiv = 1301.6967| s2cid = 9691279

}}

| 0.771(2)

Voronoi tessellation

| 0.71410(2),{{cite journal

| last = Becker

| first = A.

|author2=R. M. Ziff

| title = Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations

| journal = Physical Review E

| volume = 80

| issue = 4

| pages = 041101

| doi = 10.1103/PhysRevE.80.041101

| pmid = 19905267

| year = 2009|bibcode = 2009PhRvE..80d1101B |arxiv = 0906.4360 | s2cid = 22549508

}}

0.7151*

| 0.68,{{cite journal

| last = Shante

| first = K. S.

|author2=S. Kirkpatrick

| title = An introduction to percolation theory

| journal = Advances in Physics

| volume = 20

| issue = 85

| pages = 325–357

| doi = 10.1080/00018737100101261

| year = 1971|bibcode = 1971AdPhy..20..325S }}

0.6670(1), 0.6680(5), 0.666931(5)

Voronoi covering/medial

| 4

| 0.666931(2)

| 0.53618(2)

Randomized kagome/square-octagon, fraction r={{frac|1|2}}

| 4

| 0.6599

Penrose rhomb dual

| 4

| 0.6381(3)

| 0.5233(2)

Gabriel graph

| 4

| 0.6348(8),

{{cite journal

| last = Norrenbrock | first = C.

| title = Percolation threshold on planar Euclidean Gabriel Graphs

| journal = Journal of Physics A

| volume = 40 | issue = 31| pages = 9253–9258

| year = 2014

| arxiv = 0704.2098

| bibcode = 2007JPhA...40.9253P

| doi = 10.1088/1751-8113/40/31/005

| s2cid = 680787

}} 0.62{{cite journal

| last = Bertin

| first = E

| author2 = J.-M. Billiot |author3=R. Drouilhet

| title = Continuum percolation in the Gabriel graph

| journal = Adv. Appl. Probab.

| volume = 34

| issue = 4

| pages = 689

| year = 2002

| doi=10.1239/aap/1037990948| s2cid = 121288601

}}

| 0.5167(6), 0.52

Random-line tessellation, dual

| 4

| 0.586(2){{cite journal

| last = Lepage

| first = Thibaut

| author2=Lucie Delaby

| author3=Fausto Malvagi

| author4=Alain Mazzolo

| title =Monte Carlo simulation of fully Markovian stochastic geometries

| journal = Progress in Nuclear Science and Technology

| volume = 2

| pages = 743–748

| year = 2011

| doi=10.15669/pnst.2.743| doi-access = free

}}

Penrose rhomb

| 4

| 0.5837(3), 0.0.5610(6) (weighted bonds){{cite journal

| last = Zhang

| first = C.

|author2= K. De'Bell

| title = Reformulation of the percolation problem on a quasilattice: Estimates of the percolation threshold, chemical dimension, and amplitude ratio

| journal = Physical Review B

| volume = 47

| issue = 14

| pages = 8558–8564

| doi = 10.1103/PhysRevB.47.8558

| year = 1993| pmid = 10004894

| bibcode = 1993PhRvB..47.8558Z

}}

0.58391(1){{cite journal

| last = Ziff

| first = R. M.

|author2=F. Babalievski

| title = Site percolation on the Penrose rhomb lattice

| journal = Physica A

| volume = 269

| issue = 2–4

| pages = 201–210

| doi = 10.1016/S0378-4371(99)00166-1

| year = 1999|bibcode = 1999PhyA..269..201Z }}

| 0.483(5),{{cite journal

| last = Lu

| first = Jian Ping

|author2=Joseph L. Birman

| title = Percolation and Scaling on a Quasilattice

| journal = Journal of Statistical Physics

| volume = 46

| issue = 5/6

| pages = 1057–1066

| doi = 10.1007/BF01011156

| year = 1987| bibcode = 1987JSP....46.1057L

| s2cid = 121645524

}}

0.4770(2)

Octagonal lattice, "chemical" links (Ammann–Beenker tiling)

|0.585{{cite journal

| last = Babalievski

| first = F.

| title = Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices

| journal = Physica A

| volume = 220

| issue = 1995

| pages = 245–250

| doi = 10.1016/0378-4371(95)00260-E

| year = 1995 | bibcode = 1995PhyA..220..245B }}

|0.48

Octagonal lattice, "ferromagnetic" links

|5.17

|0.543

|0.40

Dodecagonal lattice, "chemical" links

|3.63

|0.628

|0.54

Dodecagonal lattice, "ferromagnetic" links

|4.27

|0.617

|0.495

Delaunay triangulation

| {{frac|1|2}}{{cite journal

| last = Bollobás

| first = Béla

|author2=Oliver Riordan

| title = The critical probability for random Voronoi percolation in the plane is 1/2

| journal = Probab. Theory Relat. Fields

| volume = 136

| issue = 3

| pages = 417–468

| doi = 10.1007/s00440-005-0490-z

| year = 2006| arxiv =math/0410336

| s2cid = 15985691

}}

| 0.3333(1){{cite journal

| last = Hsu

| first = H. P.

|author2=M. C. Huang

| title = Percolation thresholds, critical exponents, and scaling functions on planar random lattices and their duals

| journal = Physical Review E

| volume = 60

| issue =6

| pages = 6361–6370

| doi = 10.1103/PhysRevE.60.6361

| pmid = 11970550

| year = 1999|bibcode = 1999PhRvE..60.6361H | s2cid = 8750738

}}

0.3326(5),{{cite journal

| last = Huang

| first = Ming-Chang

|author2= Hsiao-Ping Hsu

| title = Percolation thresholds, critical exponents, and scaling functions on spherical random lattices

| journal = International Journal of Modern Physics C

| volume = 13

| issue = 3

| pages = 383–395

| doi = 10.1142/S012918310200319X

| year = 2002}}

0.333069(2)

Uniform Infinite Planar Triangulation{{cite journal

| last = Angel

| first = Omer

| author2 = Schramm, Oded

| title = Uniform infinite planar triangulation

| journal = Commun. Math. Phys.

| volume = 241

| issue = 2–3

| pages = 191–213

}}

| {{frac|1|2}}

| (2{{radic|3}} – 1)/11 ≈ 0.2240{{cite journal

| last = Bernardi

| first = Olivier

| author2 = Curien, Nicolas

| author3 = Miermont, Grėgory

| title = A Boltzmann approach to percolation on random triangulations

| journal = Canadian Journal of Mathematics

| volume = 71

| pages = 1–43

| year = 2019

| arxiv = 1705.04064| doi = 10.4153/CJM-2018-009-x

| s2cid = 6817693

}}

{{cite journal

| last = Angel

| first = O.

| author2 = Curien, Nicolas

| title = Percolations on random maps I: Half-plane models

| journal = Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

| volume = 51

| issue = 2

| pages = 405–431

| year = 2014

| doi = 10.1214/13-AIHP583 | arxiv = 1301.5311 | bibcode = 2015AIHPB..51..405A| s2cid = 14964345

}}

*Theoretical estimate

= Thresholds on 2D correlated systems =

Assuming power-law correlations $C(r) \sim |r|^{-\alpha}$

class="wikitable"

lattice

! α

! Site percolation threshold

! Bond percolation threshold

square

| 3

| 0.561406(4){{cite journal

| last = Zierenberg

| first = Johannes

| author2 = Niklas Fricke

| author3 = Martin Marenz

| author4 = F. P. Spitzner

| author5 = Viktoria Blavatska

| author6 = Wolfhard Janke

| title = Percolation thresholds and fractal dimensions for square and cubic lattices with long-range correlated defects

| journal = Physical Review E

| volume = 96

| issue = 6

| pages = 062125

| year = 2017

| doi = 10.1103/PhysRevE.96.062125 | pmid = 29347311

| arxiv = 1708.02296 | bibcode = 2017PhRvE..96f2125Z

| s2cid = 22353394

}}

square

| 2

| 0.550143(5)

square

| 0.1

| 0.508(4)

= Thresholds on slabs =

h is the thickness of the slab, h × ∞ × ∞. Boundary conditions (b.c.) refer to the top and bottom planes of the slab.

class="wikitable"
Lattice ! h ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
simple cubic (open b.c.) \|2 \|5 \|5 \| 0.47424, 0.4756{{Cite journal\|last1=Horton\|first1=M. K.\|last2=Moram\|first2=M. A.\|date=2017-04-17\|title=Alloy composition fluctuations and percolation in semiconductor alloy quantum wells\|journal=Applied Physics Letters\|language=en\|volume=110\|issue=16\|pages=162103\|doi=10.1063/1.4980089\|bibcode=2017ApPhL.110p2103H\|issn=0003-6951}} \|
bcc (open b.c.) \|2 \| \| \|0.4155 \|
hcp (open b.c.) \|2 \| \| \|0.2828 \|
diamond (open b.c.) \|2 \| \| \|0.5451 \|
simple cubic (open b.c.) \|3 \| \| \|0.4264 \|
bcc (open b.c.) \|3 \| \| \|0.3531 \|
bcc (periodic b.c.) \|3 \| \| \| \| 0.21113018(38)
hcp (open b.c.) \|3 \| \| \|0.2548 \|
diamond (open b.c.) \|3 \| \| \|0.5044 \|
simple cubic (open b.c.) \|4 \| \| \| 0.3997,{{cite journal \| last = Sotta \| first = P. \|author2=D. Long \| title = The crossover from 2D to 3D percolation: Theory and numerical simulations \| journal = Eur. Phys. J. E \| volume = 11 \| issue = 4 \| pages = 375–388 \| doi = 10.1140/epje/i2002-10161-6 \| pmid = 15011039 \| year = 2003\|bibcode = 2003EPJE...11..375S \| s2cid = 32831742 }} 0.3998 \|
bcc (open b.c.) \|4 \| \| \|0.3232 \|
bcc (periodic b.c.) \|4 \| \| \| \| 0.20235168(59)
hcp (open b.c.) \|4 \| \| \|0.2405 \|
diamond (open b.c.) \|4 \| \| \|0.4842 \|
simple cubic (periodic b.c.) \| 5 \| 6 \| 6 \| \| 0.278102(5)
simple cubic (open b.c.) \|6 \| \| \|0.3708 \|
simple cubic (periodic b.c.) \|6 \|6 \|6 \| \| 0.272380(2)
bcc (open b.c.) \|6 \| \| \|0.2948 \|
hcp (open b.c.) \|6 \| \| \|0.2261 \|
diamond (open b.c.) \|6 \| \| \|0.4642 \|
simple cubic (periodic b.c.) \| 7 \| 6 \| 6 \| 0.3459514(12){{cite journal \| last = Gliozzi \| first = F. \| author2 = S. Lottini \|author3=M. Panero \|author4=A. Rago \| title = Random percolation as a gauge theory \| journal = Nuclear Physics B \| volume = 719 \| issue = 3 \| year = 2005 \| pages = 255–274 \| doi = 10.1016/j.nuclphysb.2005.04.021\|arxiv = cond-mat/0502339 \|bibcode = 2005NuPhB.719..255G \| hdl = 2318/5995 \| s2cid = 119360708 }} \| 0.268459(1)
simple cubic (open b.c.) \|8 \| \| \|0.3557, 0.3565 \|
simple cubic (periodic b.c.) \|8 \|6 \|6 \| \|0.265615(5)
bcc (open b.c.) \|8 \| \| \|0.2811 \|
hcp (open b.c.) \|8 \| \| \|0.2190 \|
diamond (open b.c.) \|8 \| \| \|0.4549 \|
simple cubic (open b.c.) \|12 \| \| \|0.3411 \|
bcc (open b.c.) \|12 \| \| \|0.2688 \|
hcp (open b.c.) \|12 \| \| \|0.2117 \|
diamond (open b.c.) \|12 \| \| \|0.4456 \|
simple cubic (open b.c.) \|16 \| \| \|0.3219, 0.3339 \|
bcc (open b.c.) \|16 \| \| \|0.2622 \|
hcp (open b.c.) \|16 \| \| \|0.2086 \|
diamond (open b.c.) \|16 \| \| \|0.4415 \|
simple cubic (open b.c.) \|32 \| \| \|0.3219, \|
simple cubic (open b.c.) \|64 \| \| \|0.3165, \|
simple cubic (open b.c.) \|128 \| \| \|0.31398, \|

class="wikitable"

Lattice

! h

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

simple cubic (open b.c.)

| 0.47424, 0.4756{{Cite journal|last1=Horton|first1=M. K.|last2=Moram|first2=M. A.|date=2017-04-17|title=Alloy composition fluctuations and percolation in semiconductor alloy quantum wells|journal=Applied Physics Letters|language=en|volume=110|issue=16|pages=162103|doi=10.1063/1.4980089|bibcode=2017ApPhL.110p2103H|issn=0003-6951}}

bcc (open b.c.)

|0.4155

hcp (open b.c.)

|0.2828

diamond (open b.c.)

|0.5451

simple cubic (open b.c.)

|0.4264

bcc (open b.c.)

|0.3531

bcc (periodic b.c.)

| 0.21113018(38)

hcp (open b.c.)

|0.2548

diamond (open b.c.)

|0.5044

simple cubic (open b.c.)

| 0.3997,{{cite journal

| last = Sotta

| first = P.

|author2=D. Long

| title = The crossover from 2D to 3D percolation: Theory and numerical simulations

| journal = Eur. Phys. J. E

| volume = 11

| issue = 4

| pages = 375–388

| doi = 10.1140/epje/i2002-10161-6

| pmid = 15011039

| year = 2003|bibcode = 2003EPJE...11..375S | s2cid = 32831742

}}

0.3998

bcc (open b.c.)

|0.3232

bcc (periodic b.c.)

| 0.20235168(59)

hcp (open b.c.)

|0.2405

diamond (open b.c.)

|0.4842

simple cubic (periodic b.c.)

| 5

| 6

| 0.278102(5)

simple cubic (open b.c.)

|0.3708

simple cubic (periodic b.c.)

| 0.272380(2)

bcc (open b.c.)

|0.2948

hcp (open b.c.)

|0.2261

diamond (open b.c.)

|0.4642

simple cubic (periodic b.c.)

| 7

| 6

| 0.3459514(12){{cite journal

| last = Gliozzi

| first = F.

| author2 = S. Lottini |author3=M. Panero |author4=A. Rago

| title = Random percolation as a gauge theory

| journal = Nuclear Physics B

| volume = 719

| issue = 3

| year = 2005

| pages = 255–274

| doi = 10.1016/j.nuclphysb.2005.04.021|arxiv = cond-mat/0502339 |bibcode = 2005NuPhB.719..255G | hdl = 2318/5995

| s2cid = 119360708

}}

| 0.268459(1)

simple cubic (open b.c.)

|0.3557, 0.3565

simple cubic (periodic b.c.)

|0.265615(5)

bcc (open b.c.)

|0.2811

hcp (open b.c.)

|0.2190

diamond (open b.c.)

|0.4549

simple cubic (open b.c.)

|12

|0.3411

bcc (open b.c.)

|12

|0.2688

hcp (open b.c.)

|12

|0.2117

diamond (open b.c.)

|12

|0.4456

simple cubic (open b.c.)

|16

|0.3219, 0.3339

bcc (open b.c.)

|16

|0.2622

hcp (open b.c.)

|16

|0.2086

diamond (open b.c.)

|16

|0.4415

simple cubic (open b.c.)

|32

|0.3219,

simple cubic (open b.c.)

|64

|0.3165,

simple cubic (open b.c.)

|128

|0.31398,

Percolation in 3D

class="wikitable"
Lattice ! z ! $\overline z$ ! filling factor* ! filling fraction* ! width=41% \| Site percolation threshold ! width=25% \| Bond percolation threshold
(10,3)-a oxide (or site-bond) {{cite journal \| last = Yoo \| first = Ted Y. \|author2=Jonathan Tran \|author3=Shane P. Stahlheber \|author4=Carina E. Kaainoa \|author5=Kevin Djepang \|author6=Alexander R. Small \| title = Site percolation on lattices with low average coordination numbers \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2014 \| issue = 6 \| year = 2014 \| pages = P06014 \| arxiv = 1403.1676\|bibcode = 2014JSMTE..06..014Y \| doi=10.1088/1742-5468/2014/06/p06014\| s2cid = 119290405 }} \| 2³ 3² \| 2.4 \| \| \| 0.748713(22) \| = (p_c,bond(10,3) – a)^{{frac\|1\|2}} = 0.742334(25)
(10,3)-b oxide (or site-bond) \| 2³ 3² \| 2.4 \| 0.233 {{cite journal \| last = Wells \| first = A. F. \| title = Structures Based on the 3-Connected Net 10³ – b \| journal = Journal of Solid State Chemistry \| volume = 54 \| issue = 3 \| year = 1984 \| pages = 378–388 \| doi = 10.1016/0022-4596(84)90169-5\| bibcode = 1984JSSCh..54..378W}} \| 0.174 \| 0.745317(25) \| = (p_c,bond(10,3) – b)^{{frac\|1\|2}} = 0.739388(22)
silicon dioxide (diamond site-bond) \| 4,2² \| 2 {{frac\|2\|3}} \| \| \| 0.638683(35) \|
Modified (10,3)-b \| 3²,2 \| 2 {{frac\|2\|3}} \| \| \| \| 0.627 {{cite journal \| last = Pant \| first = Mihir \|author2=Don Towsley \|author3=Dirk Englund \|author4=Saikat Guha \| title = Percolation thresholds for photonic quantum computing \| journal = Nature Communications \| year = 2017 \| volume = 10 \| issue = 1 \| pages = 1070 \| doi = 10.1038/s41467-019-08948-x \| pmid = 30842425 \| pmc = 6403388 \| arxiv = 1701.03775 }}
(8,3)-a \| 3 \| 3 \| \| \| 0.577962(33){{cite journal \| last = Tran \| first = Jonathan \|author2=Ted Yoo \|author3=Shane Stahlheber \|author4=Alex Small \| title = Percolation thresholds on 3-dimensional lattices with 3 nearest neighbors \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2013 \| issue = 5 \| year = 2013 \| pages = P05014 \| arxiv = 1211.6531\|bibcode = 2013JSMTE..05..014T \|doi = 10.1088/1742-5468/2013/05/P05014 \| s2cid = 119182062 }} \| 0.555700(22)
(10,3)-a gyroid{{cite journal \| last = Hyde \| first = Stephen T. \|author2= O'Keeffe, Michael \|author3=Proserpio, Davide M. \| title = A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics \| journal = Angew. Chem. Int. Ed. \| volume = 47 \| issue = 42 \| year = 2008 \| pages = 7996–8000 \| doi = 10.1002/anie.200801519 \| pmid=18767088}} \| 3 \| 3 \| \| \| 0.571404(40) \| 0.551060(37)
(10,3)-b \| 3 \| 3 \| \| \| 0.565442(40) \| 0.546694(33)
cubic oxide (cubic site-bond) \| 6,2³ \| 3.5 \| \| \| 0.524652(50) \|
bcc dual \| \| 4 \| \| \| 0.4560(6) \| 0.4031(6)
ice Ih \| 4 \| 4 \| π {{radic\|3}} / 16 = 0.340087 \| 0.147 \| 0.433(11){{cite journal \| last = Frisch \| first = H. L. \| author2 = E. Sonnenblick \|author3=V. A. Vyssotsky \|author4=J. M. Hammersley \| title = Critical Percolation Probabilities (Site Problem) \| journal = Physical Review \| volume = 124 \| issue = 4 \| year = 1961 \| pages = 1021–1022 \| doi = 10.1103/PhysRev.124.1021\|bibcode = 1961PhRv..124.1021F }} \| 0.388(10){{cite journal \| last = Vyssotsky \| first = V. A. \| author2 = S. B. Gordon \|author3=H. L. Frisch \|author4=J. M. Hammersley \| title = Critical Percolation Probabilities (Bond Problem) \| journal = Physical Review \| volume = 123 \| issue = 5 \| year = 1961 \| pages = 1566–1567 \| doi = 10.1103/PhysRev.123.1566\|bibcode = 1961PhRv..123.1566V }}
diamond (Ice Ic) \| 4 \| 4 \|π {{radic\|3}} / 16 = 0.340087 \| 0.1462332 \| 0.4299(8), 0.4299870(4),{{cite journal \| last = Xu \| first = Xiao \| author2 = Junfeng Wang \|author3=Jian-Ping Lv \|author4=Youjin Deng \| title = Simultaneous analysis of three-dimensional percolation models \| journal = Frontiers of Physics \| volume = 9 \| issue = 1 \| year = 2014 \| pages = 113–119 \| doi = 10.1007/s11467-013-0403-z \| arxiv = 1310.5399 \|bibcode = 2014FrPhy...9..113X \| s2cid = 119250232 }} {{val\|0.426\|0.08\|0.02}},{{cite journal \| last = Silverman \| first = Amihal \|author2=J. Adler\|author2-link= Joan Adler \| title = Site-percolation threshold for a diamond lattice with diatomic substitution \| journal = Physical Review B \| volume = 42 \| issue = 2 \| year = 1990 \| pages = 1369–1373 \| doi = 10.1103/PhysRevB.42.1369\| pmid = 9995550 \|bibcode = 1990PhRvB..42.1369S }} 0.4297(4) {{cite journal \| last = van der Marck \| first = Steven C. \| title = Erratum: Percolation thresholds and universal formulas [Phys. Rev. E 55, 1514 (1997)] \| journal = Physical Review E \| volume = 56 \| issue = 3 \| year = 1997 \| page = 3732 \| doi = 10.1103/PhysRevE.56.3732.2 \| bibcode = 1997PhRvE..56.3732V \| doi-access = free }} 0.4301(4),{{cite journal \| last = van der Marck \| first = Steven C. \| title = Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices \| journal = International Journal of Modern Physics C \| volume = 9 \| issue = 4 \| year = 1998 \| pages = 529–540 \| doi = 10.1142/S0129183198000431 \|arxiv = cond-mat/9802187 \|bibcode = 1998IJMPC...9..529V \| s2cid = 119097158 }} 0.428(4), 0.425(15),{{cite journal \| last = Sykes \| first = M. F. \| author2=J. W. Essam \| title = Critical percolation probabilities by series method \| journal = Physical Review \| volume = 133 \| issue = 1A \| year = 1964 \| pages = A310–A315 \| doi = 10.1103/PhysRev.133.A310 \| bibcode = 1964PhRv..133..310S }} 0.425, 0.436(12) \| 0.3895892(5), 0.3893(2), 0.3893(3), 0.388(5), 0.3886(5), 0.388(5){{cite journal \| last = Sykes \| first = M. F. \| author2 = D. S. Gaunt \| author3 = M. Glen \| title = Percolation processes in three dimensions \| journal = J. Phys. A: Math. Gen. \| volume = 9 \| issue = 10 \| year = 1976 \| pages = 1705–1712 \| doi = 10.1088/0305-4470/9/10/021 \| bibcode = 1976JPhA....9.1705S }} 0.390(11)
diamond dual \| \| 6 {{frac\|2\|3}} \| \| \|0.3904(5) {{cite journal \| last = van der Marck \| first = Steven C. \| title = Percolation thresholds of the duals of the face-centered-cubic, hexagonal-close-packed, and diamond lattices \| journal = Physical Review E \| volume = 55 \| issue = 6 \| year = 1997 \| pages = 6593–6597 \| doi = 10.1103/PhysRevE.55.6593\| bibcode = 1997PhRvE..55.6593V}} \| 0.2350(5)
3D kagome (covering graph of the diamond lattice) \|6 \| \|π {{radic\|2}} / 12 = 0.37024 \| 0.1442 \|0.3895(2) =p_c(site) for diamond dual and p_c(bond) for diamond lattice \| 0.2709(6)
Bow-tie stack dual \| \| 5 {{frac\|1\|3}} \| \| \| 0.3480(4) \| 0.2853(4)
honeycomb stack \| 5 \| 5 \| \| \| 0.3701(2) \| 0.3093(2)
octagonal stack dual \| 5 \| 5 \| \| \| 0.3840(4) \| 0.3168(4)
pentagonal stack \| \| 5 {{frac\|1\|3}} \| \| \| 0.3394(4) \| 0.2793(4)
kagome stack \| 6 \| 6 \| 0.453450 \| 0.1517 \| 0.3346(4) \| 0.2563(2)
fcc dual \| 4²,8 \| 5 {{frac\|1\|3}} \| \| \| 0.3341(5) \| 0.2703(3)
simple cubic \| 6 \| 6 \| π / 6 = 0.5235988 \| 0.1631574 \| 0.307(10), 0.307, 0.3115(5),{{cite journal \| last = Sur \| first = Amit \|author2=Joel L. Lebowitz \|author3=J. Marro \|author4=M. H. Kalos\|author5=S. Kirkpatrick \| title = Monte Carlo studies of percolation phenomena for a simple cubic lattice \| journal = Journal of Statistical Physics \| volume = 15 \| issue = 5 \| year = 1976 \| pages = 345–353 \| doi = 10.1007/BF01020338 \|bibcode=1976JSP....15..345S\| s2cid = 38734613 }} 0.3116077(2),{{cite journal \| last = Wang \| first = J \|author2=Z. Zhou \|author3=W. Zhang \|author4=T. Garoni \|author5=Y. Deng \| title = Bond and site percolation in three dimensions \| journal = Physical Review E \| volume = 87 \| issue = 5 \| year = 2013 \| pages = 052107 \| arxiv = 1302.0421\|bibcode = 2013PhRvE..87e2107W \|doi = 10.1103/PhysRevE.87.052107 \| pmid = 23767487 \| s2cid = 14087496 }} 0.311604(6),{{cite journal \| last = Grassberger \| first = P. \| title = Numerical studies of critical percolation in three dimensions \| journal = J. Phys. A \| volume = 25 \| issue = 22 \| year = 1992 \| pages = 5867–5888 \| doi = 10.1088/0305-4470/25/22/015\|bibcode = 1992JPhA...25.5867G }} 0.311605(5),{{cite journal \| last = Acharyya \| first = M. \|author2=D. Stauffer \| title = Effects of Boundary Conditions on the Critical Spanning Probability \| journal = Int. J. Mod. Phys. C \| volume = 9 \| issue = 4 \| year = 1998 \| pages = 643–647 \| doi = 10.1142/S0129183198000534\|arxiv = cond-mat/9805355 \|bibcode = 1998IJMPC...9..643A \| s2cid = 15684907 }} 0.311600(5),{{cite journal \| last = Jan \| first = N. \|author2=D. Stauffer \| title = Random Site Percolation in Three Dimensions \| journal = Int. J. Mod. Phys. C \| volume = 9 \| issue = 4 \| year = 1998 \| pages = 341–347 \| doi = 10.1142/S0129183198000261\|bibcode = 1998IJMPC...9..341J }} 0.3116077(4),{{cite journal \| last = Deng \| first = Youjin \|author2=H. W. J. Blöte \| title = Monte Carlo study of the site-percolation model in two and three dimensions \| journal = Physical Review E \| volume = 72 \| issue = 1 \| year = 2005 \| pages = 016126 \| doi = 10.1103/PhysRevE.72.016126\| pmid = 16090055 \|bibcode = 2005PhRvE..72a6126D \| url = http://resolver.tudelft.nl/uuid:e987a69f-6dde-4d8d-a616-bfcde4d5bdac }} 0.3116081(13),{{cite journal \| last = Ballesteros \| first = P. N. \| author2 = L. A. Fernández \|author3=V. Martín-Mayor \|author4=A. Muñoz Sudepe \|author5=G. Parisi \|author6=J. J. Ruiz-Lorenzo \| title = Scaling corrections: site percolation and Ising model in three dimensions \| journal = Journal of Physics A \| volume = 32 \| issue = 1 \| year = 1999 \| pages = 1–13 \| doi = 10.1088/0305-4470/32/1/004\|arxiv = cond-mat/9805125 \|bibcode = 1999JPhA...32....1B \| s2cid = 2787294 }} 0.3116080(4), 0.3116060(48), 0.3116004(35),{{cite journal \| last = Škvor \| first = Jiří \|author2=Ivo Nezbeda \| title = Percolation threshold parameters of fluids \| journal = Physical Review E \| volume = 79 \| issue = 4 \| year = 2009 \| pages = 041141 \| doi = 10.1103/PhysRevE.79.041141\| pmid = 19518207 \|bibcode = 2009PhRvE..79d1141S }} 0.31160768(15) \| 0.247(5), 0.2479(4), 0.2488(2),{{cite journal \| last = Adler \| first = Joan \| author2 = Yigal Meir \| author3 = Amnon Aharony \| author4 = A. B. Harris \| author5 = Lior Klein \| title = Low-Concentration Series in General Dimension \| journal = Journal of Statistical Physics \| volume = 58 \| issue = 3/4 \| year = 1990 \| pages = 511–538 \| doi = 10.1007/BF01112760\| bibcode = 1990JSP....58..511A \| s2cid = 122109020 }} 0.24881182(10), 0.2488125(25), 0.2488126(5),{{cite journal \| last = Lorenz \| first = C. D. \|author2=R. M. Ziff \| title = Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices \| journal = Physical Review E \| volume = 57 \| issue = 1 \| year = 1998 \| pages = 230–236 \| doi = 10.1103/PhysRevE.57.230\|arxiv = cond-mat/9710044 \|bibcode = 1998PhRvE..57..230L \| s2cid = 119074750 }}
hcp dual \| 4⁴,8² \| 5 {{frac\|1\|3}} \| \| \| 0.3101(5) \| 0.2573(3)
dice stack \| 5,8 \| 6 \| π {{radic\|3}} / 9 = 0.604600 \| 0.1813 \| 0.2998(4) \| 0.2378(4)
bow-tie stack \| 7 \| 7 \| \| \| 0.2822(6) \| 0.2092(4)
Stacked triangular / simple hexagonal \| 8 \| 8 \| \| \| 0.26240(5),{{cite journal \| last = Schrenk \| first = K. J. \|author2=N. A. M. Araújo \|author3=H. J. Herrmann \| title = Stacked triangular lattice: percolation properties \| journal = Physical Review E \| volume = 87 \| issue = 3 \| year = 2013 \| pages = 032123 \| arxiv = 1302.0484 \| doi = 10.1103/PhysRevE.87.032123\|bibcode = 2013PhRvE..87c2123S \| s2cid = 2917074 }} 0.2625(2),{{cite journal \| last = Martins \| first = P. \|author2=J. Plascak \| title = Percolation on two- and three-dimensional lattices \| journal = Physical Review \| volume = 67 \| issue = 4 \| year = 2003 \| pages = 046119 \|bibcode = 2003PhRvE..67d6119M \| doi=10.1103/physreve.67.046119\| pmid = 12786448 \| arxiv =cond-mat/0304024\| s2cid = 31891392 }} 0.2623(2) \| 0.18602(2), 0.1859(2)
octagonal (union-jack) stack \| 6,10 \| 8 \| \| \| 0.2524(6) \| 0.1752(2)
bcc \| 8 \| 8 \| \| \| 0.243(10), 0.243, 0.2459615(10), 0.2460(3),{{cite journal \| last = Bradley \| first = R. M. \| author2 = P. N. Strenski \|author3=J.-M. Debierre \| title = Surfaces of percolation clusters in three dimensions \| journal = Physical Review B \| volume = 44 \| issue = 1 \| year = 1991 \| pages = 76–84 \| doi = 10.1103/PhysRevB.44.76\| pmid = 9998221 \|bibcode = 1991PhRvB..44...76B }} 0.2464(7),{{cite journal \| last = Gaunt \| first = D. S. \|author2=M. F. Sykes \| title = Series study of random percolation in three dimensions \| journal = J. Phys. A \| volume = 16 \| issue = 4 \| year = 1983 \| pages = 783 \| doi = 10.1088/0305-4470/16/4/016\|bibcode = 1983JPhA...16..783G }} 0.2458(2) \| 0.178(5), 0.1795(3), 0.18025(15), 0.1802875(10)
simple cubic with 3NN (same as bcc) \| 8 \| 8 \| \| \| 0.2455(1), 0.2457(7){{cite journal \| last = Gallyamov \| first = S. R. \| author2 = S.A. Melchukov \| title = Percolation threshold of a simple cubic lattice with fourth neighbors: the theory and numerical calculation with parallelization \| journal = Third International Conference "High Performance Computing" HPC-UA 2013 (Ukraine, Kyiv, October 7–11, 2013) \| year = 2013 \| url = http://hpc-ua.org/hpc-ua-13/files/proceedings/24.pdf \| access-date = August 23, 2019 \| archive-date = August 23, 2019 \| archive-url = https://web.archive.org/web/20190823170156/http://hpc-ua.org/hpc-ua-13/files/proceedings/24.pdf \| url-status = dead }} \|
fcc, D₃ \| 12 \| 12 \| π / (3 {{radic\|2}}) = 0.740480 \| 0.147530 \|0.195, 0.198(3),{{cite journal \| last = Sykes \| first = M. F. \|author2=D. S. Gaunt \| author3 = J. W. Essam \| title = The percolation probability for the site problem on the face-centred cubic lattice \| journal = Journal of Physics A \| volume = 9 \| issue = 5 \| year = 1976 \| pages = L43–L46 \| doi = 10.1088/0305-4470/9/5/002\|bibcode = 1976JPhA....9L..43S }} 0.1998(6), 0.1992365(10),{{cite journal \| last = Lorenz \| first = C. D. \|author2=R. M. Ziff \| title = Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation \| journal = Journal of Physics A \| volume = 31 \| issue = 40 \| year = 1998 \| pages = 8147–8157 \| doi = 10.1088/0305-4470/31/40/009\|arxiv = cond-mat/9806224 \|bibcode = 1998JPhA...31.8147L \| s2cid = 12493873 }} 0.19923517(20), 0.1994(2), 0.199236(4){{cite journal \| last = Hu \| first = Yi \|author2 = Patrick Charbonneau \| title = Percolation thresholds on high-dimensional D_n and E₈-related lattices \| journal = Physical Review E \| volume = 103 \| issue = 6 \| year = 2021 \| pages = 062115 \| doi = 10.1103/PhysRevE.103.062115 \| pmid = 34271715 \| arxiv = 2102.09682 \| bibcode = 2021PhRvE.103f2115H \| s2cid = 231979212 }} \| 0.1198(3), 0.1201635(10) 0.120169(2)
hcp \| 12 \| 12 \| π / (3 {{radic\|2}}) = 0.740480 \| 0.147545 \| 0.195(5), 0.1992555(10){{cite journal \| last = Lorenz \| first = C. D. \|author2=R. May \|author3=R. M. Ziff \| title = Similarity of Percolation Thresholds on the HCP and FCC Lattices \| journal = Journal of Statistical Physics \| volume = 98 \| issue = 3/4 \| year = 2000 \| pages = 961–970 \| doi = 10.1023/A:1018648130343\| hdl = 2027.42/45178 \| s2cid = 10950378 \| url = https://deepblue.lib.umich.edu/bitstream/2027.42/45178/1/10955_2004_Article_222436.pdf \| hdl-access = free }} \| 0.1201640(10), 0.119(2)
La_2−x Sr_x Cu O₄ \| 12 \| 12 \| \| \| 0.19927(2){{cite journal \| last = Tahir-Kheli \| first = Jamil \|author2=W. A. Goddard III \|title = Chiral plaquette polaron theory of cuprate superconductivity \| journal = Physical Review B \| volume = 76 \| issue = 1 \| year = 2007 \| pages = 014514 \|doi = 10.1103/PhysRevB.76.014514\|bibcode = 2007PhRvB..76a4514T \|arxiv = 0707.3535 \| s2cid = 8882419 }} \|
simple cubic with 2NN (same as fcc) \| 12 \| 12 \| \| \| 0.1991(1) \|
simple cubic with NN+4NN \| 12 \| 12 \| \| \| 0.15040(12),{{cite journal \| last = Malarz \| first = Krzysztof \| title = Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors \| journal = Physical Review E \| volume = 91 \| issue = 4 \| date = 2015 \| pages = 043301 \| doi = 10.1103/PhysRevE.91.043301 \| pmid = 25974606 \| bibcode = 2015PhRvE..91d3301M \| arxiv = 1501.01586\| s2cid = 37943657 }} 0.1503793(7) \| 0.1068263(7){{cite journal \| last = Xun \| first = Zhipeng \| author2 = Robert M. Ziff \| title = Bond percolation on simple cubic lattices with extended neighborhoods \| journal = Physical Review E \| volume = 102 \| issue = 4 \| date = 2020 \| pages = 012102 \| doi = 10.1103/PhysRevE.102.012102 \| pmid = 32795057 \| arxiv = 2001.00349\| bibcode = 2020PhRvE.102a2102X \| s2cid = 209531616 }}
simple cubic with 3NN+4NN \| 14 \| 14 \| \| \| 0.20490(12) \| 0.1012133(7)
bcc NN+2NN (= sc(3,4) sc-3NN+4NN) \| 14 \| 14 \| \| \| 0.175, 0.1686,(20) 0.1759432(8) \| 0.0991(5), 0.1012133(7),{{cite journal \| last = Xun \| first = Zhipeng \| author2 = DaPeng Hao \| author3 = Robert M. Ziff \| title = Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions \| journal = Physical Review E \| volume = 105 \| issue = 2 \| year = 2022 \| pages = 024105 \| doi = 10.1103/PhysRevE.105.024105 \| pmid = 35291074 \| arxiv = 2111.10975 \| bibcode = 2022PhRvE.105b4105X \| s2cid = 244478657 }} 0.1759432(8)
Nanotube fibers on FCC \| 14 \| 14 \| \| \| 0.1533(13){{cite journal \| last = Xu \| first = Fangbo \|author2 = Zhiping Xu \|author3 = Boris I. Yakobson \|title = Site-Percolation Threshold of Carbon Nanotube Fibers---Fast Inspection of Percolation with Markov Stochastic Theory \| journal = Physica A \| volume = 407 \| year = 2014 \| pages = 341–349 \|doi = 10.1016/j.physa.2014.04.013 \|arxiv = 1401.2130\| bibcode = 2014PhyA..407..341X \| s2cid = 119267606 }} \|
simple cubic with NN+3NN \| 14 \| 14 \| \| \| 0.1420(1){{cite journal \| last = Kurzawski \| first = Ł. \|author2=K. Malarz \| title = Simple cubic random-site percolation thresholds for complex neighbourhoods \| journal = Rep. Math. Phys. \| volume = 70 \| issue = 2 \| year = 2012 \| pages = 163–169 \| doi = 10.1016/S0034-4877(12)60036-6 \| bibcode = 2012RpMP...70..163K \| arxiv = 1111.3254\| citeseerx = 10.1.1.743.1726 \| s2cid = 119120046 }} \| 0.0920213(7)
simple cubic with 2NN+4NN \| 18 \| 18 \| \| \| 0.15950(12) \|0.0751589(9)
simple cubic with NN+2NN \| 18 \| 18 \| \| \| 0.137, 0.136, 0.1372(1), 0.13735(5),{{citation needed\|date=June 2019}} 0.1373045(5) \| 0.0752326(6)
fcc with NN+2NN (=sc-2NN+4NN) \| 18 \| 18 \| \| \| 0.136, 0.1361408(8) \| 0.0751589(9)
simple cubic with short-length correlation \| 6+ \| 6+ \| \| \| 0.126(1){{cite journal \| last = Harter \| first = T. \| title = Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields \| journal = Physical Review E \| volume = 72 \| issue =2 \| year = 2005 \| pages = 026120 \| doi =10.1103/PhysRevE.72.026120 \| pmid = 16196657 \|bibcode = 2005PhRvE..72b6120H \| s2cid = 2708506 }} \|
simple cubic with NN+3NN+4NN \| 20 \| 20 \| \| \| 0.11920(12) \| 0.0624379(9)
simple cubic with 2NN+3NN \| 20 \| 20 \| \| \| 0.1036(1) \| 0.0629283(7)
simple cubic with NN+2NN+4NN \| 24 \| 24 \| \| \| 0.11440(12) \| 0.0533056(6)
simple cubic with 2NN+3NN+4NN \| 26 \| 26 \| \| \| 0.11330(12) \| 0.0474609(9)
simple cubic with NN+2NN+3NN \| 26 \| 26 \| \| \| 0.097, 0.0976(1), 0.0976445(10), 0.0976444(6) \| 0.0497080(10)
bcc with NN+2NN+3NN \| 26 \| 26 \| \| \| 0.095, 0.0959084(6) \| 0.0492760(10)
simple cubic with NN+2NN+3NN+4NN \| 32 \| 32 \| \| \| 0.10000(12), 0.0801171(9) \| 0.0392312(8)
fcc with NN+2NN+3NN \| 42 \| 42 \| \| \| 0.061, 0.0610(5),{{cite journal \| last = Gawron \| first = T. R. \| author2 = Marek Cieplak \| title = Site percolation thresholds of the FCC lattice \| journal = Acta Physica Polonica A \| volume = 80 \| issue = 3 \| year = 1991 \| pages = 461 \| doi = 10.12693/APhysPolA.80.461 \| bibcode = 1991AcPPA..80..461G \| url = http://przyrbwn.icm.edu.pl/APP/PDF/80/a080z3p38.pdf \| doi-access = free }} 0.0618842(8) \| 0.0290193(7)
fcc with NN+2NN+3NN+4NN \| 54 \| 54 \| \| \| 0.0500(5) \|
sc-1,2,3,4,5 simple cubic with NN+2NN+3NN+4NN+5NN \| 56 \| 56 \| \| \| 0.0461815(5) \| 0.0210977(7)
sc-1,...,6 (2x2x2 cube ) \| 80 \| 80 \| \| \| 0.0337049(9), 0.03373(13) \| 0.0143950(10)
sc-1,...,7 \| 92 \| 92 \| \| \| 0.0290800(10) \| 0.0123632(8)
sc-1,...,8 \| 122 \| 122 \| \| \| 0.0218686(6) \| 0.0091337(7)
sc-1,...,9 \| 146 \| 146 \| \| \| 0.0184060(10) \| 0.0075532(8)
sc-1,...,10 \| 170 \| 170 \| \| \| \| 0.0064352(8)
sc-1,...,11 \| 178 \| 178 \| \| \| \| 0.0061312(8)
sc-1,...,12 \| 202 \| 202 \| \| \| \| 0.0053670(10)
sc-1,...,13 \| 250 \| 250 \| \| \| \| 0.0042962(8)
3x3x3 cube \| 274 \| 274 \| \| \| φ_c= 0.76564(1), p_c = 0.0098417(7), 0.009854(6) \|
4x4x4 cube \| 636 \| 636 \| \| \| φ_c=0.76362(1), p_c = 0.0042050(2), 0.004217(3) \|
5x5x5 cube \| 1214 \| 1250 \| \| \| φ_c=0.76044(2), p_c = 0.0021885(2), 0.002185(4) \|
6x6x6 cube \| 2056 \| 2056 \| \| \| 0.001289(2) \|

class="wikitable"

Lattice

! z

! $\overline z$

! filling factor*

! filling fraction*

! width=41% | Site percolation threshold

! width=25% | Bond percolation threshold

(10,3)-a oxide (or site-bond)

{{cite journal

| last = Yoo

| first = Ted Y.

| title = Site percolation on lattices with low average coordination numbers

| journal = Journal of Statistical Mechanics: Theory and Experiment

| volume = 2014

| issue = 6

| year = 2014

| pages = P06014

| arxiv = 1403.1676|bibcode = 2014JSMTE..06..014Y | doi=10.1088/1742-5468/2014/06/p06014| s2cid = 119290405

}}

| 2³ 3²

| 2.4

| 0.748713(22)

| = (p_c,bond(10,3) – a)^{{frac|1|2}} = 0.742334(25)

(10,3)-b oxide (or site-bond)

| 2³ 3²

| 2.4

| 0.233

{{cite journal

| last = Wells

| first = A. F.

| title = Structures Based on the 3-Connected Net 10³ – b

| journal = Journal of Solid State Chemistry

| volume = 54

| issue = 3

| year = 1984

| pages = 378–388

| doi = 10.1016/0022-4596(84)90169-5| bibcode = 1984JSSCh..54..378W}}

| 0.174

| 0.745317(25)

| = (p_c,bond(10,3) – b)^{{frac|1|2}} = 0.739388(22)

silicon dioxide (diamond site-bond)

| 4,2²

| 2 {{frac|2|3}}

| 0.638683(35)

Modified (10,3)-b

| 3²,2

| 2 {{frac|2|3}}

| 0.627

{{cite journal

| last = Pant

| first = Mihir

|author2=Don Towsley |author3=Dirk Englund |author4=Saikat Guha

| title = Percolation thresholds for photonic quantum computing

| journal = Nature Communications

| year = 2017

| volume = 10

| issue = 1

| pages = 1070

| doi = 10.1038/s41467-019-08948-x

| pmid = 30842425

| pmc = 6403388

| arxiv = 1701.03775

}}

(8,3)-a

| 3

| 0.577962(33){{cite journal

| last = Tran

| first = Jonathan

|author2=Ted Yoo |author3=Shane Stahlheber |author4=Alex Small

| title = Percolation thresholds on 3-dimensional lattices with 3 nearest neighbors

| journal = Journal of Statistical Mechanics: Theory and Experiment

| volume = 2013

| issue = 5

| year = 2013

| pages = P05014

| arxiv = 1211.6531|bibcode = 2013JSMTE..05..014T |doi = 10.1088/1742-5468/2013/05/P05014 | s2cid = 119182062

}}

| 0.555700(22)

(10,3)-a gyroid{{cite journal

| last = Hyde

| first = Stephen T.

|author2= O'Keeffe, Michael |author3=Proserpio, Davide M.

| title = A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics

| journal = Angew. Chem. Int. Ed.

| volume = 47

| issue = 42

| year = 2008

| pages = 7996–8000

| doi = 10.1002/anie.200801519

| pmid=18767088}}

| 3

| 0.571404(40)

| 0.551060(37)

(10,3)-b

| 3

| 0.565442(40)

| 0.546694(33)

cubic oxide (cubic site-bond)

| 6,2³

| 3.5

| 0.524652(50)

bcc dual

| 4

| 0.4560(6)

| 0.4031(6)

ice Ih

| 4

| π {{radic|3}} / 16 = 0.340087

| 0.147

| 0.433(11){{cite journal

| last = Frisch

| first = H. L.

| author2 = E. Sonnenblick |author3=V. A. Vyssotsky |author4=J. M. Hammersley

| title = Critical Percolation Probabilities (Site Problem)

| journal = Physical Review

| volume = 124

| issue = 4

| year = 1961

| pages = 1021–1022

| doi = 10.1103/PhysRev.124.1021|bibcode = 1961PhRv..124.1021F }}

| 0.388(10){{cite journal

| last = Vyssotsky

| first = V. A.

| author2 = S. B. Gordon |author3=H. L. Frisch |author4=J. M. Hammersley

| title = Critical Percolation Probabilities (Bond Problem)

| journal = Physical Review

| volume = 123

| issue = 5

| year = 1961

| pages = 1566–1567

| doi = 10.1103/PhysRev.123.1566|bibcode = 1961PhRv..123.1566V }}

diamond (Ice Ic)

| 4

|π {{radic|3}} / 16 = 0.340087

| 0.1462332

| 0.4299(8), 0.4299870(4),{{cite journal

| last = Xu

| first = Xiao

| author2 = Junfeng Wang |author3=Jian-Ping Lv |author4=Youjin Deng

| title = Simultaneous analysis of three-dimensional percolation models

| journal = Frontiers of Physics

| volume = 9

| issue = 1

| year = 2014

| pages = 113–119

| doi = 10.1007/s11467-013-0403-z

| arxiv = 1310.5399 |bibcode = 2014FrPhy...9..113X | s2cid = 119250232

}} {{val|0.426|0.08|0.02}},{{cite journal

| last = Silverman

| first = Amihal

|author2=J. Adler|author2-link= Joan Adler

| title = Site-percolation threshold for a diamond lattice with diatomic substitution

| journal = Physical Review B

| volume = 42

| issue = 2

| year = 1990

| pages = 1369–1373

| doi = 10.1103/PhysRevB.42.1369| pmid = 9995550

|bibcode = 1990PhRvB..42.1369S }}

0.4297(4) {{cite journal

| last = van der Marck

| first = Steven C.

| title = Erratum: Percolation thresholds and universal formulas [Phys. Rev. E 55, 1514 (1997)]

| journal = Physical Review E

| volume = 56

| issue = 3

| year = 1997

| page = 3732

| doi = 10.1103/PhysRevE.56.3732.2

| bibcode = 1997PhRvE..56.3732V

| doi-access = free

}} 0.4301(4),{{cite journal

| last = van der Marck

| first = Steven C.

| title = Calculation of Percolation Thresholds in High Dimensions for FCC, BCC and Diamond Lattices

| journal = International Journal of Modern Physics C

| volume = 9

| issue = 4

| year = 1998

| pages = 529–540

| doi = 10.1142/S0129183198000431

|arxiv = cond-mat/9802187

|bibcode = 1998IJMPC...9..529V

| s2cid = 119097158

}} 0.428(4), 0.425(15),{{cite journal

| last = Sykes

| first = M. F.

| author2=J. W. Essam

| title = Critical percolation probabilities by series method

| journal = Physical Review

| volume = 133

| issue = 1A

| year = 1964

| pages = A310–A315

| doi = 10.1103/PhysRev.133.A310

| bibcode = 1964PhRv..133..310S

}} 0.425, 0.436(12)

| 0.3895892(5), 0.3893(2), 0.3893(3), 0.388(5), 0.3886(5), 0.388(5){{cite journal

| last = Sykes

| first = M. F.

| author2 = D. S. Gaunt

| author3 = M. Glen

| title = Percolation processes in three dimensions

| journal = J. Phys. A: Math. Gen.

| volume = 9

| issue = 10

| year = 1976

| pages = 1705–1712

| doi = 10.1088/0305-4470/9/10/021

| bibcode = 1976JPhA....9.1705S

}} 0.390(11)

diamond dual

| 6 {{frac|2|3}}

|0.3904(5)

{{cite journal

| last = van der Marck

| first = Steven C.

| title = Percolation thresholds of the duals of the face-centered-cubic, hexagonal-close-packed, and diamond lattices

| journal = Physical Review E

| volume = 55

| issue = 6

| year = 1997

| pages = 6593–6597

| doi = 10.1103/PhysRevE.55.6593| bibcode = 1997PhRvE..55.6593V}}

| 0.2350(5)

3D kagome (covering graph of the diamond lattice)

|π {{radic|2}} / 12 = 0.37024

| 0.1442

|0.3895(2) =p_c(site) for diamond dual and p_c(bond) for diamond lattice

| 0.2709(6)

Bow-tie stack dual

| 5 {{frac|1|3}}

| 0.3480(4)

| 0.2853(4)

honeycomb stack

| 5

| 0.3701(2)

| 0.3093(2)

octagonal stack dual

| 5

| 0.3840(4)

| 0.3168(4)

pentagonal stack

| 5 {{frac|1|3}}

| 0.3394(4)

| 0.2793(4)

kagome stack

| 6

| 0.453450

| 0.1517

| 0.3346(4)

| 0.2563(2)

fcc dual

| 4²,8

| 5 {{frac|1|3}}

| 0.3341(5)

| 0.2703(3)

simple cubic

| 6

| π / 6 = 0.5235988

| 0.1631574

| 0.307(10), 0.307, 0.3115(5),{{cite journal

| last = Sur

| first = Amit

|author2=Joel L. Lebowitz |author3=J. Marro |author4=M. H. Kalos|author5=S. Kirkpatrick

| title = Monte Carlo studies of percolation phenomena for a simple cubic lattice

| journal = Journal of Statistical Physics

| volume = 15

| issue = 5

| year = 1976

| pages = 345–353

| doi = 10.1007/BF01020338 |bibcode=1976JSP....15..345S| s2cid = 38734613

}} 0.3116077(2),{{cite journal

| last = Wang

| first = J

|author2=Z. Zhou |author3=W. Zhang |author4=T. Garoni |author5=Y. Deng

| title = Bond and site percolation in three dimensions

| journal = Physical Review E

| volume = 87

| issue = 5

| year = 2013

| pages = 052107

| arxiv = 1302.0421|bibcode = 2013PhRvE..87e2107W |doi = 10.1103/PhysRevE.87.052107 | pmid = 23767487

| s2cid = 14087496

}} 0.311604(6),{{cite journal

| last = Grassberger

| first = P.

| title = Numerical studies of critical percolation in three dimensions

| journal = J. Phys. A

| volume = 25

| issue = 22

| year = 1992

| pages = 5867–5888

| doi = 10.1088/0305-4470/25/22/015|bibcode = 1992JPhA...25.5867G }}

0.311605(5),{{cite journal

| last = Acharyya

| first = M.

|author2=D. Stauffer

| title = Effects of Boundary Conditions on the Critical Spanning Probability

| journal = Int. J. Mod. Phys. C

| volume = 9

| issue = 4

| year = 1998

| pages = 643–647

| doi = 10.1142/S0129183198000534|arxiv = cond-mat/9805355 |bibcode = 1998IJMPC...9..643A | s2cid = 15684907

}}

0.311600(5),{{cite journal

| last = Jan

| first = N.

|author2=D. Stauffer

| title = Random Site Percolation in Three Dimensions

| journal = Int. J. Mod. Phys. C

| volume = 9

| issue = 4

| year = 1998

| pages = 341–347

| doi = 10.1142/S0129183198000261|bibcode = 1998IJMPC...9..341J }}

0.3116077(4),{{cite journal

| last = Deng

| first = Youjin

|author2=H. W. J. Blöte

| title = Monte Carlo study of the site-percolation model in two and three dimensions

| journal = Physical Review E

| volume = 72

| issue = 1

| year = 2005

| pages = 016126

| doi = 10.1103/PhysRevE.72.016126| pmid = 16090055

|bibcode = 2005PhRvE..72a6126D | url = http://resolver.tudelft.nl/uuid:e987a69f-6dde-4d8d-a616-bfcde4d5bdac

}}

0.3116081(13),{{cite journal

| last = Ballesteros

| first = P. N.

| title = Scaling corrections: site percolation and Ising model in three dimensions

| journal = Journal of Physics A

| volume = 32

| issue = 1

| year = 1999

| pages = 1–13

| doi = 10.1088/0305-4470/32/1/004|arxiv = cond-mat/9805125 |bibcode = 1999JPhA...32....1B | s2cid = 2787294

}}

0.3116080(4), 0.3116060(48), 0.3116004(35),{{cite journal

| last = Škvor

| first = Jiří

|author2=Ivo Nezbeda

| title = Percolation threshold parameters of fluids

| journal = Physical Review E

| volume = 79

| issue = 4

| year = 2009

| pages = 041141

| doi = 10.1103/PhysRevE.79.041141| pmid = 19518207

|bibcode = 2009PhRvE..79d1141S }}

0.31160768(15)

| 0.247(5), 0.2479(4), 0.2488(2),{{cite journal

| last = Adler

| first = Joan

| author2 = Yigal Meir

| author3 = Amnon Aharony

| author4 = A. B. Harris

| author5 = Lior Klein

| title = Low-Concentration Series in General Dimension

| journal = Journal of Statistical Physics

| volume = 58

| issue = 3/4

| year = 1990

| pages = 511–538

| doi = 10.1007/BF01112760| bibcode = 1990JSP....58..511A

| s2cid = 122109020

}} 0.24881182(10), 0.2488125(25), 0.2488126(5),{{cite journal

| last = Lorenz

| first = C. D.

|author2=R. M. Ziff

| title = Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices

| journal = Physical Review E

| volume = 57

| issue = 1

| year = 1998

| pages = 230–236

| doi = 10.1103/PhysRevE.57.230|arxiv = cond-mat/9710044 |bibcode = 1998PhRvE..57..230L | s2cid = 119074750

}}

hcp dual

| 4⁴,8²

| 5 {{frac|1|3}}

| 0.3101(5)

| 0.2573(3)

dice stack

| 5,8

| 6

| π {{radic|3}} / 9 = 0.604600

| 0.1813

| 0.2998(4)

| 0.2378(4)

bow-tie stack

| 7

| 0.2822(6)

| 0.2092(4)

Stacked triangular / simple hexagonal

| 8

| 0.26240(5),{{cite journal

| last = Schrenk

| first = K. J.

|author2=N. A. M. Araújo |author3=H. J. Herrmann

| title = Stacked triangular lattice: percolation properties

| journal = Physical Review E

| volume = 87

| issue = 3

| year = 2013

| pages = 032123

| arxiv = 1302.0484

| doi = 10.1103/PhysRevE.87.032123|bibcode = 2013PhRvE..87c2123S | s2cid = 2917074

}} 0.2625(2),{{cite journal

| last = Martins

| first = P.

|author2=J. Plascak

| title = Percolation on two- and three-dimensional lattices

| journal = Physical Review

| volume = 67

| issue = 4

| year = 2003

| pages = 046119

|bibcode = 2003PhRvE..67d6119M

| doi=10.1103/physreve.67.046119| pmid = 12786448

| arxiv =cond-mat/0304024| s2cid = 31891392

}}

0.2623(2)

| 0.18602(2), 0.1859(2)

octagonal (union-jack) stack

| 6,10

| 8

| 0.2524(6)

| 0.1752(2)

bcc

| 8

| 0.243(10), 0.243, 0.2459615(10), 0.2460(3),{{cite journal

| last = Bradley

| first = R. M.

| author2 = P. N. Strenski |author3=J.-M. Debierre

| title = Surfaces of percolation clusters in three dimensions

| journal = Physical Review B

| volume = 44

| issue = 1

| year = 1991

| pages = 76–84

| doi = 10.1103/PhysRevB.44.76| pmid = 9998221

|bibcode = 1991PhRvB..44...76B }}

0.2464(7),{{cite journal

| last = Gaunt

| first = D. S.

|author2=M. F. Sykes

| title = Series study of random percolation in three dimensions

| journal = J. Phys. A

| volume = 16

| issue = 4

| year = 1983

| pages = 783

| doi = 10.1088/0305-4470/16/4/016|bibcode = 1983JPhA...16..783G }}

0.2458(2)

| 0.178(5), 0.1795(3), 0.18025(15), 0.1802875(10)

simple cubic with 3NN (same as bcc)

| 8

| 0.2455(1), 0.2457(7){{cite journal

| last = Gallyamov

| first = S. R.

| author2 = S.A. Melchukov

| title = Percolation threshold of a simple cubic lattice with fourth neighbors: the theory and numerical calculation with parallelization

| journal = Third International Conference "High Performance Computing" HPC-UA 2013 (Ukraine, Kyiv, October 7–11, 2013)

| year = 2013

| url = http://hpc-ua.org/hpc-ua-13/files/proceedings/24.pdf

| access-date = August 23, 2019

| archive-date = August 23, 2019

| archive-url = https://web.archive.org/web/20190823170156/http://hpc-ua.org/hpc-ua-13/files/proceedings/24.pdf

| url-status = dead

}}

fcc, D₃

| 12

| π / (3 {{radic|2}}) = 0.740480

| 0.147530

|0.195, 0.198(3),{{cite journal

| last = Sykes

| first = M. F.

|author2=D. S. Gaunt

| author3 = J. W. Essam

| title = The percolation probability for the site problem on the face-centred cubic lattice

| journal = Journal of Physics A

| volume = 9

| issue = 5

| year = 1976

| pages = L43–L46

| doi = 10.1088/0305-4470/9/5/002|bibcode = 1976JPhA....9L..43S }}

0.1998(6), 0.1992365(10),{{cite journal

| last = Lorenz

| first = C. D.

|author2=R. M. Ziff

| title = Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation

| journal = Journal of Physics A

| volume = 31

| issue = 40

| year = 1998

| pages = 8147–8157

| doi = 10.1088/0305-4470/31/40/009|arxiv = cond-mat/9806224 |bibcode = 1998JPhA...31.8147L | s2cid = 12493873

}}

0.19923517(20), 0.1994(2), 0.199236(4){{cite journal

| last = Hu

| first = Yi

|author2 = Patrick Charbonneau

| title = Percolation thresholds on high-dimensional D_n and E₈-related lattices

| journal = Physical Review E

| volume = 103

| issue = 6

| year = 2021

| pages = 062115

| doi = 10.1103/PhysRevE.103.062115 | pmid = 34271715

| arxiv = 2102.09682

| bibcode = 2021PhRvE.103f2115H

| s2cid = 231979212

}}

| 0.1198(3), 0.1201635(10) 0.120169(2)

hcp

| 12

| π / (3 {{radic|2}}) = 0.740480

| 0.147545

| 0.195(5), 0.1992555(10){{cite journal

| last = Lorenz

| first = C. D.

|author2=R. May |author3=R. M. Ziff

| title = Similarity of Percolation Thresholds on the HCP and FCC Lattices

| journal = Journal of Statistical Physics

| volume = 98

| issue = 3/4

| year = 2000

| pages = 961–970

| doi = 10.1023/A:1018648130343| hdl = 2027.42/45178

| s2cid = 10950378

| url = https://deepblue.lib.umich.edu/bitstream/2027.42/45178/1/10955_2004_Article_222436.pdf

| hdl-access = free

}}

| 0.1201640(10), 0.119(2)

La_2−x Sr_x Cu O₄

| 12

| 0.19927(2){{cite journal

| last = Tahir-Kheli

| first = Jamil

|author2=W. A. Goddard III

|title = Chiral plaquette polaron theory of cuprate superconductivity

| journal = Physical Review B

| volume = 76

| issue = 1

| year = 2007

| pages = 014514

|doi = 10.1103/PhysRevB.76.014514|bibcode = 2007PhRvB..76a4514T |arxiv = 0707.3535 | s2cid = 8882419

}}

simple cubic with 2NN (same as fcc)

| 12

| 0.1991(1)

simple cubic with NN+4NN

| 12

| 0.15040(12),{{cite journal

| last = Malarz

| first = Krzysztof

| title = Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

| journal = Physical Review E

| volume = 91

| issue = 4

| date = 2015

| pages = 043301

| doi = 10.1103/PhysRevE.91.043301

| pmid = 25974606

| bibcode = 2015PhRvE..91d3301M

| arxiv = 1501.01586| s2cid = 37943657

}}

0.1503793(7)

| 0.1068263(7){{cite journal

| last = Xun

| first = Zhipeng

| author2 = Robert M. Ziff

| title = Bond percolation on simple cubic lattices with extended neighborhoods

| journal = Physical Review E

| volume = 102

| issue = 4

| date = 2020

| pages = 012102

| doi = 10.1103/PhysRevE.102.012102

| pmid = 32795057

| arxiv = 2001.00349| bibcode = 2020PhRvE.102a2102X

| s2cid = 209531616

}}

simple cubic with 3NN+4NN

| 14

| 0.20490(12)

| 0.1012133(7)

bcc NN+2NN (= sc(3,4) sc-3NN+4NN)

| 14

| 0.175, 0.1686,(20) 0.1759432(8)

| 0.0991(5), 0.1012133(7),{{cite journal

| last = Xun

| first = Zhipeng

| author2 = DaPeng Hao

| author3 = Robert M. Ziff

| title = Site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in two and three dimensions

| journal = Physical Review E

| volume = 105

| issue = 2

| year = 2022

| pages = 024105

| doi = 10.1103/PhysRevE.105.024105

| pmid = 35291074

| arxiv = 2111.10975 | bibcode = 2022PhRvE.105b4105X

| s2cid = 244478657

}} 0.1759432(8)

Nanotube fibers on FCC

| 14

| 0.1533(13){{cite journal

| last = Xu

| first = Fangbo

|author2 = Zhiping Xu

|author3 = Boris I. Yakobson

|title = Site-Percolation Threshold of Carbon Nanotube Fibers---Fast Inspection of Percolation with Markov Stochastic Theory

| journal = Physica A

| volume = 407

| year = 2014

| pages = 341–349

|doi = 10.1016/j.physa.2014.04.013 |arxiv = 1401.2130| bibcode = 2014PhyA..407..341X

| s2cid = 119267606

}}

simple cubic with NN+3NN

| 14

| 0.1420(1){{cite journal

| last = Kurzawski

| first = Ł.

|author2=K. Malarz

| title = Simple cubic random-site percolation thresholds for complex neighbourhoods

| journal = Rep. Math. Phys.

| volume = 70

| issue = 2

| year = 2012

| pages = 163–169

| doi = 10.1016/S0034-4877(12)60036-6

| bibcode = 2012RpMP...70..163K

| arxiv = 1111.3254| citeseerx = 10.1.1.743.1726

| s2cid = 119120046

}}

| 0.0920213(7)

simple cubic with 2NN+4NN

| 18

| 0.15950(12)

|0.0751589(9)

simple cubic with NN+2NN

| 18

| 0.137, 0.136, 0.1372(1), 0.13735(5),{{citation needed|date=June 2019}} 0.1373045(5)

| 0.0752326(6)

fcc with NN+2NN (=sc-2NN+4NN)

| 18

| 0.136, 0.1361408(8)

| 0.0751589(9)

simple cubic with short-length correlation

| 6+

| 0.126(1){{cite journal

| last = Harter

| first = T.

| title = Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields

| journal = Physical Review E

| volume = 72

| issue =2

| year = 2005

| pages = 026120

| doi =10.1103/PhysRevE.72.026120 | pmid = 16196657

|bibcode = 2005PhRvE..72b6120H | s2cid = 2708506

}}

simple cubic with NN+3NN+4NN

| 20

| 0.11920(12)

| 0.0624379(9)

simple cubic with 2NN+3NN

| 20

| 0.1036(1)

| 0.0629283(7)

simple cubic with NN+2NN+4NN

| 24

| 0.11440(12)

| 0.0533056(6)

simple cubic with 2NN+3NN+4NN

| 26

| 0.11330(12)

| 0.0474609(9)

simple cubic with NN+2NN+3NN

| 26

| 0.097, 0.0976(1), 0.0976445(10), 0.0976444(6)

| 0.0497080(10)

bcc with NN+2NN+3NN

| 26

| 0.095, 0.0959084(6)

| 0.0492760(10)

simple cubic with NN+2NN+3NN+4NN

| 32

| 0.10000(12), 0.0801171(9)

| 0.0392312(8)

fcc with NN+2NN+3NN

| 42

| 0.061, 0.0610(5),{{cite journal

| last = Gawron

| first = T. R.

| author2 = Marek Cieplak

| title = Site percolation thresholds of the FCC lattice

| journal = Acta Physica Polonica A

| volume = 80

| issue = 3

| year = 1991

| pages = 461

| doi = 10.12693/APhysPolA.80.461

| bibcode = 1991AcPPA..80..461G

| url = http://przyrbwn.icm.edu.pl/APP/PDF/80/a080z3p38.pdf

| doi-access = free

}} 0.0618842(8)

| 0.0290193(7)

fcc with NN+2NN+3NN+4NN

| 54

| 0.0500(5)

sc-1,2,3,4,5 simple cubic with NN+2NN+3NN+4NN+5NN

| 56

| 0.0461815(5)

| 0.0210977(7)

sc-1,...,6 (2x2x2 cube )

| 80

| 0.0337049(9), 0.03373(13)

| 0.0143950(10)

sc-1,...,7

| 92

| 0.0290800(10)

| 0.0123632(8)

sc-1,...,8

| 122

| 0.0218686(6)

| 0.0091337(7)

sc-1,...,9

| 146

| 0.0184060(10)

| 0.0075532(8)

sc-1,...,10

| 170

| 0.0064352(8)

sc-1,...,11

| 178

| 0.0061312(8)

sc-1,...,12

| 202

| 0.0053670(10)

sc-1,...,13

| 250

| 0.0042962(8)

3x3x3 cube

| 274

| φ_c= 0.76564(1), p_c = 0.0098417(7), 0.009854(6)

4x4x4 cube

| 636

| φ_c=0.76362(1), p_c = 0.0042050(2), 0.004217(3)

5x5x5 cube

| 1214

| 1250

| φ_c=0.76044(2), p_c = 0.0021885(2), 0.002185(4)

6x6x6 cube

| 2056

| 0.001289(2)

Filling factor = fraction of space filled by touching spheres at every lattice site (for systems with uniform bond length only). Also called Atomic Packing Factor.

Filling fraction (or Critical Filling Fraction) = filling factor * p_c(site).

NN = nearest neighbor, 2NN = next-nearest neighbor, 3NN = next-next-nearest neighbor, etc.

kxkxk cubes are cubes of occupied sites on a lattice, and are equivalent to extended-range percolation of a cube of length (2k+1), with edges and corners removed, with z = (2k+1)³-12(2k-1)-9 (center site not counted in z).

Question: the bond thresholds for the hcp and fcc lattice

agree within the small statistical error. Are they identical,

and if not, how far apart are they? Which threshold is expected to be bigger? Similarly for the ice and diamond lattices. See {{cite journal

| last1=Sykes | first1=M. F.

| last2=Rehr | first2=J. J.

| last3=Glen | first3=Maureen

| title = A note on the percolation probabilities of pairs of closely similar lattices

| journal = Mathematical Proceedings of the Cambridge Philosophical Society

| volume = 76

| year = 1996

| pages = 389–392

| doi =10.1017/S0305004100049021| s2cid = 96528423

}}

class="wikitable"

! System

! polymer Φ_c

percolating excluded volume of athermal polymer matrix (bond-fluctuation model on cubic lattice)

| 0.4304(3){{cite journal

| last = Weber

| first = H.

|author2=W. Paul

| title = Penetrant diffusion in frozen polymer matrices: A finite-size scaling study of free volume percolation

| journal = Physical Review E

| volume = 54

| issue = 4

| year = 1996

| pages = 3999–4007

| doi = 10.1103/PhysRevE.54.3999 | pmid = 9965547

|bibcode = 1996PhRvE..54.3999W }}

= 3D distorted lattices =

Here, one distorts a regular lattice of unit spacing by moving vertices uniformly within the cube $(x-\alpha,x+\alpha),(y-\alpha,y+\alpha),(z-\alpha,z+\alpha)$ , and considers percolation when sites are within Euclidean distance $d$ of each other.

class="wikitable"

! Lattice

! $\overline z$

! $\alpha$

! $d$

! Site percolation threshold

! Bond percolation threshold

cubic

| 0.05

| 1.0

| 0.60254(3)

| 0.1

| 1.00625

| 0.58688(4)

| 0.15

| 1.025

| 0.55075(2)

| 0.175

| 1.05

| 0.50645(5)

| 0.2

| 1.1

| 0.44342(3)

= Overlapping shapes on 3D lattices =

Site threshold is the number of overlapping objects per lattice site. The coverage φ_c is the net fraction of sites covered, and v is the volume (number of cubes). Overlapping cubes are given in the section on thresholds of 3D lattices. Here z is the coordination number to k-mers of either orientation, with $z=6k^2+18k-4$

class="wikitable"
System ! k ! z ! Site coverage φ_c ! Site percolation threshold p_c
1 x 2 dimer, cubic lattice \| 2 \| 56 \| 0.24542 \| 0.045847(2)
1 x 3 trimer, cubic lattice \| 3 \| 104 \| 0.19578 \| 0.023919(9)
1 x 4 stick, cubic lattice \| 4 \| 164 \| 0.16055 \| 0.014478(7)
1 x 5 stick, cubic lattice \| 5 \| 236 \| 0.13488 \| 0.009613(8)
1 x 6 stick, cubic lattice \| 6 \| 320 \| 0.11569 \| 0.006807(2)
2 x 2 plaquette, cubic lattice \| 2 \| \| 0.22710 \| 0.021238(2)
3 x 3 plaquette, cubic lattice \| 3 \| \| 0.18686 \| 0.007632(5)
4 x 4 plaquette, cubic lattice \| 4 \| \| 0.16159 \| 0.003665(3)
5 x 5 plaquette, cubic lattice \| 5 \| \| 0.14316 \| 0.002058(5)
6 x 6 plaquette, cubic lattice \| 6 \| \| 0.12900 \| 0.001278(5)

class="wikitable"

System

! k

! z

! Site coverage φ_c

! Site percolation threshold p_c

1 x 2 dimer, cubic lattice

| 2

| 56

| 0.24542

| 0.045847(2)

1 x 3 trimer, cubic lattice

| 3

| 104

| 0.19578

| 0.023919(9)

1 x 4 stick, cubic lattice

| 4

| 164

| 0.16055

| 0.014478(7)

1 x 5 stick, cubic lattice

| 5

| 236

| 0.13488

| 0.009613(8)

1 x 6 stick, cubic lattice

| 6

| 320

| 0.11569

| 0.006807(2)

2 x 2 plaquette, cubic lattice

| 2

| 0.22710

| 0.021238(2)

3 x 3 plaquette, cubic lattice

| 3

| 0.18686

| 0.007632(5)

4 x 4 plaquette, cubic lattice

| 4

| 0.16159

| 0.003665(3)

5 x 5 plaquette, cubic lattice

| 5

| 0.14316

| 0.002058(5)

6 x 6 plaquette, cubic lattice

| 6

| 0.12900

| 0.001278(5)

The coverage is calculated from $p_c$ by $\phi_c = 1-(1-p_c)^{3 k}$ for sticks, and $\phi_c = 1-(1-p_c)^{3 k^2}$ for plaquettes.

= Dimer percolation in 3D =

class="wikitable"
System ! Site percolation threshold ! Bond percolation threshold
Simple cubic \| \| 0.2555(1)

class="wikitable"

System

! Site percolation threshold

! Bond percolation threshold

Simple cubic

| 0.2555(1)

= Thresholds for 3D continuum models =

All overlapping except for jammed spheres and polymer matrix.

class="wikitable"
System ! Φ_c ! η_c
Spheres of radius r \| 0.289,{{cite journal \| last = Holcomb \| first = D F.. \| author2= J. J. Rehr, Jr. \| title = Percolation in heavily doped semiconductors* \| journal = Physical Review \| volume = 183 \| issue = 3 \| year = 1969 \| pages = 773–776 \| doi = 10.1103/PhysRev.183.773\| bibcode = 1969PhRv..183..773H }} 0.293,{{cite journal \| last = Holcomb \| first = D F. \| author2= F. Holcomb \|author3 = M. Iwasawa \| title = Clustering of randomly placed spheres \| journal = Biometrika \| volume = 59 \| year = 1972 \| pages = 207–209 \| doi = 10.1093/biomet/59.1.207}} 0.286,{{cite journal \| last = Shante \| first = Vinod K. S. \|author2= Scott Kirkpatrick \| title = An introduction to percolation theory \| journal = Advances in Physics \| volume = 20 \| issue = 85 \| year = 1971 \| pages = 325–357 \| doi = 10.1080/00018737100101261\| bibcode = 1971AdPhy..20..325S }} 0.295. 0.2895(5),{{cite journal \| last = Rintoul \| first = M. D. \|author2= S. Torquato \| title = Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model \| journal = J. Phys. A: Math. Gen. \| volume = 30 \| issue = 16 \| year = 1997 \| pages = L585 \| doi = 10.1088/0305-4470/30/16/005\|bibcode = 1997JPhA...30L.585R \| citeseerx = 10.1.1.42.4284}} 0.28955(7),{{cite journal \| last = Consiglio \| first = R. \|author2=R. Baker \|author3=G. Paul \|author4=H. E. Stanley \| title = Continuum percolation of congruent overlapping spherocylinders \| journal = Physica A \| volume = 319 \| year = 2003 \| pages = 49–55 \| doi = 10.1016/S0378-4371(02)01501-7 }} 0.2896(7),{{cite journal \| last = Xu \| first = Wenxiang \|author2=Xianglong Su \|author3=Yang Jiao \| title = Continuum percolation of congruent overlapping spherocylinders \| journal = Physical Review E \| volume = 93 \| issue = 3 \| year = 2016 \| pages = 032122 \| doi = 10.1103/PhysRevE.94.032122 \| pmid = 27078307 \| bibcode = 2016PhRvE..94c2122X }} 0.289573(2),{{cite journal \| last = Lorenz \| first = C. D. \|author2=R. M. Ziff \| title = Precise determination of the critical percolation threshold for the three dimensional Swiss cheese model using a growth algorithm \| journal = J. Chem. Phys. \| volume = 114 \| issue = 8 \| year = 2000 \| pages = 3659 \| doi = 10.1063/1.1338506 \|bibcode = 2001JChPh.114.3659L \| url = https://deepblue.lib.umich.edu/bitstream/2027.42/70114/2/JCPSA6-114-8-3659-1.pdf \| hdl = 2027.42/70114 \| hdl-access = free }} 0.2896,{{cite journal \|last1=Lin \|first1=Jianjun \|last2=Chen \|first2=Huisu \|last3=Xu \|first3=Wenxiang \|title=Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems \|journal=Physical Review E \|date=2018 \|volume=98 \|issue=1 \|page=012134 \|doi=10.1103/PhysRevE.98.012134\|pmid=30110832 \|bibcode=2018PhRvE..98a2134L \|s2cid=52017287}} 0.2854, 0.290,{{Cite journal\|last1=Li \|first1=Mingqi\|last2=Chen\|first2=Huisu\|last3=Lin\|first3=Jianjun\|date=January 2020\|title=Efficient measurement of the percolation threshold for random systems of congruent overlapping ovoids\|journal=Powder Technology\|volume=360\|pages=598–607\|doi=10.1016/j.powtec.2019.10.044\|s2cid=208693526\|issn=0032-5910}} 0.290,{{Cite journal\|last1=Li \|first1=Mingqi\|last2=Chen\|first2=Huisu\|last3=Lin\|first3=Jianjun \|date=April 2020\|title=Numerical study for the percolation threshold and transport properties of porous composites comprising non-centrosymmetrical superovoidal pores \|journal=Computer Methods in Applied Mechanics and Engineering\|volume=361\|pages=112815\|doi=10.1016/j.cma.2019.112815\|bibcode=2020CMAME.361k2815L \|s2cid=213152892\|issn=0045-7825}} 0.2895693(26){{Cite journal \|last=Brzeski \|first=Piotr \|last2=Kondrat \|first2=Grzegorz \|date=2022 \|title=Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous \|url=https://iopscience.iop.org/article/10.1088/1742-5468/ac6519 \|journal=Journal of Statistical Mechanics: Theory and Experiment \|volume=2022 \|issue=5 \|pages=053202 \|doi=10.1088/1742-5468/ac6519 \|issn=1742-5468}} \| 0.3418(7), 0.3438(13), 0.341889(3), 0.3360, 0.34189(2) [corrected], 0.341935(8), 0.335,
Oblate ellipsoids with major radius r and aspect ratio {{frac\|4\|3}} \| 0.2831{{cite journal \| last = Garboczi \| first = E. J. \|author2=K. A. Snyder \|author3=J. F. Douglas \| title = Geometrical percolation threshold of overlapping ellipsoids \| journal = Physical Review E \| volume = 52 \| issue = 1 \| year = 1995 \| pages = 819–827 \|bibcode = 1995PhRvE..52..819G \|doi = 10.1103/PhysRevE.52.819 \| pmid = 9963485 \| url = https://zenodo.org/record/1233793 }} \| 0.3328
Prolate ellipsoids with minor radius r and aspect ratio {{frac\|3\|2}} \| 0.2757, 0.2795, 0.2763 \| 0.3278
Oblate ellipsoids with major radius r and aspect ratio 2 \| 0.2537, 0.2629, 0.254 \| 0.3050
Prolate ellipsoids with minor radius r and aspect ratio 2 \| 0.2537, 0.2618, 0.25(2),{{cite journal \| last = Yi \| first = Y.-B. \|author2=A. M. Sastry \| title = Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution \| journal = Proc. R. Soc. Lond. A \| volume =460 \| issue =2048 \| year = 2004 \| pages =2353–2380 \| doi = 10.1098/rspa.2004.1279 \|bibcode = 2004RSPSA.460.2353Y \| s2cid = 2475482 }} 0.2507 \| 0.3035, 0.29(3)
Oblate ellipsoids with major radius r and aspect ratio 3 \| 0.2289 \| 0.2599
Prolate ellipsoids with minor radius r and aspect ratio 3 \| 0.2033, 0.2244, 0.20(2) \| 0.2541, 0.22(3)
Oblate ellipsoids with major radius r and aspect ratio 4 \| 0.2003 \| 0.2235
Prolate ellipsoids with minor radius r and aspect ratio 4 \| 0.1901, 0.16(2) \| 0.2108, 0.17(3)
Oblate ellipsoids with major radius r and aspect ratio 5 \| 0.1757 \| 0.1932
Prolate ellipsoids with minor radius r and aspect ratio 5 \| 0.1627, 0.13(2) \| 0.1776, 0.15(2)
Oblate ellipsoids with major radius r and aspect ratio 10 \| 0.0895, 0.1058 \| 0.1118
Prolate ellipsoids with minor radius r and aspect ratio 10 \| 0.0724, 0.08703, 0.07(2) \| 0.09105, 0.07(2)
Oblate ellipsoids with major radius r and aspect ratio 100 \| 0.01248 \| 0.01256
Prolate ellipsoids with minor radius r and aspect ratio 100 \| 0.006949 \| 0.006973
Oblate ellipsoids with major radius r and aspect ratio 1000 \| 0.001275 \| 0.001276
Oblate ellipsoids with major radius r and aspect ratio 2000 \| 0.000637 \| 0.000637
Spherocylinders with H/D = 1 \| 0.2439(2) \|
Spherocylinders with H/D = 4 \| 0.1345(1) \|
Spherocylinders with H/D = 10 \| 0.06418(20) \|
Spherocylinders with H/D = 50 \| 0.01440(8) \|
Spherocylinders with H/D = 100 \| 0.007156(50) \|
Spherocylinders with H/D = 200 \| 0.003724(90) \|
Aligned cylinders \| 0.2819(2) \| 0.3312(1){{cite journal \| last = Hyytiä \| first = E. \| author2 = J. Virtamo \|author3=P. Lassila \|author4=J. Ott \| title = Continuum percolation threshold for permeable aligned cylinders and opportunistic networking \| journal = IEEE Communications Letters \| volume = 16 \| issue = 7 \| year = 2012 \| pages = 1064–1067 \| doi = 10.1109/LCOMM.2012.051512.120497 \| s2cid = 1056865 \| url = https://zenodo.org/record/1223952 }}
Aligned cubes of side $\ell = 2 a$ \| 0.2773(2) 0.27727(2), 0.27730261(79) \| 0.3247(3), 0.3248(3), 0.32476(4) 0.324766(1)
Randomly oriented icosahedra \| \| 0.3030(5)
Randomly oriented dodecahedra \| \| 0.2949(5)
Randomly oriented octahedra \| \| 0.2514(6)
Randomly oriented cubes of side $\ell = 2 a$ \| 0.2168(2) 0.2174, \| 0.2444(3), 0.2443(5){{cite journal \| last = Torquato \| first = S. \|author2=Y. Jiao \| title = Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles \| journal = Physical Review E \| volume = 87 \| issue = 2 \| year = 2012 \| arxiv = 1210.0134 \| pages = 022111 \| doi = 10.1103/PhysRevE.87.022111\| pmid = 23496464 \|bibcode = 2013PhRvE..87b2111T \| s2cid = 11417012 }}
Randomly oriented tetrahedra \| \| 0.1701(7)
Randomly oriented disks of radius r (in 3D) \| \| 0.9614(5){{cite journal \| last = Yi \| first = Y. B. \|author2=E. Tawerghi \| title = Geometric percolation thresholds of interpenetrating plates in three-dimensional space \| journal = Physical Review E \| volume = 79 \| issue = 4 \| year = 2009 \| pages = 041134 \| doi = 10.1103/PhysRevE.79.041134 \| pmid = 19518200 \|bibcode = 2009PhRvE..79d1134Y }}
Randomly oriented square plates of side $\sqrt{\pi} r$ \| \| 0.8647(6)
Randomly oriented triangular plates of side $\sqrt{2 \pi} /3^{1/4} r$ \| \| 0.7295(6)
Jammed spheres (average z = 6) \| 0.183(3), 0.1990, {{cite journal \| last = Ziff \| first = R. M. \|author2=Salvatore Torquato \| title = Percolation of disordered jammed sphere packings \| journal = Journal of Physics A: Mathematical and Theoretical \| volume =50 \| issue = 8 \| year = 2016 \| pages = 085001 \| arxiv = 1611.00279\| bibcode =2017JPhA...50h5001Z\| doi =10.1088/1751-8121/aa5664\| s2cid = 53003822 }} see also contact network of jammed spheres below. \| 0.59(1) (volume fraction of all spheres)

class="wikitable"

System

! Φ_c

! η_c

Spheres of radius r

| 0.289,{{cite journal

| last = Holcomb

| first = D F..

| author2= J. J. Rehr, Jr.

| title = Percolation in heavily doped semiconductors*

| journal = Physical Review

| volume = 183

| issue = 3

| year = 1969

| pages = 773–776

| doi = 10.1103/PhysRev.183.773| bibcode = 1969PhRv..183..773H

}}

0.293,{{cite journal

| last = Holcomb

| first = D F.

| author2= F. Holcomb

|author3 = M. Iwasawa

| title = Clustering of randomly placed spheres

| journal = Biometrika

| volume = 59

| year = 1972

| pages = 207–209

| doi = 10.1093/biomet/59.1.207}}

0.286,{{cite journal

| last = Shante

| first = Vinod K. S.

|author2= Scott Kirkpatrick

| title = An introduction to percolation theory

| journal = Advances in Physics

| volume = 20

| issue = 85

| year = 1971

| pages = 325–357

| doi = 10.1080/00018737100101261| bibcode = 1971AdPhy..20..325S

}}

0.295. 0.2895(5),{{cite journal

| last = Rintoul

| first = M. D.

|author2= S. Torquato

| title = Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model

| journal = J. Phys. A: Math. Gen.

| volume = 30

| issue = 16

| year = 1997

| pages = L585

| doi = 10.1088/0305-4470/30/16/005|bibcode = 1997JPhA...30L.585R | citeseerx = 10.1.1.42.4284}}

0.28955(7),{{cite journal

| last = Consiglio

| first = R.

|author2=R. Baker

|author3=G. Paul

|author4=H. E. Stanley

| title = Continuum percolation of congruent overlapping spherocylinders

| journal = Physica A

| volume = 319

| year = 2003

| pages = 49–55

| doi = 10.1016/S0378-4371(02)01501-7

}}

0.2896(7),{{cite journal

| last = Xu

| first = Wenxiang

|author2=Xianglong Su

|author3=Yang Jiao

| title = Continuum percolation of congruent overlapping spherocylinders

| journal = Physical Review E

| volume = 93

| issue = 3

| year = 2016

| pages = 032122

| doi = 10.1103/PhysRevE.94.032122

| pmid = 27078307

| bibcode = 2016PhRvE..94c2122X

}}

0.289573(2),{{cite journal

| last = Lorenz

| first = C. D.

|author2=R. M. Ziff

| title = Precise determination of the critical percolation threshold for the three dimensional Swiss cheese model using a growth algorithm

| journal = J. Chem. Phys.

| volume = 114

| issue = 8

| year = 2000

| pages = 3659

| doi = 10.1063/1.1338506 |bibcode = 2001JChPh.114.3659L | url = https://deepblue.lib.umich.edu/bitstream/2027.42/70114/2/JCPSA6-114-8-3659-1.pdf

| hdl = 2027.42/70114

| hdl-access = free

}}

0.2896,{{cite journal |last1=Lin |first1=Jianjun |last2=Chen |first2=Huisu |last3=Xu |first3=Wenxiang |title=Geometrical percolation threshold of congruent cuboidlike particles in overlapping particle systems |journal=Physical Review E |date=2018 |volume=98 |issue=1 |page=012134 |doi=10.1103/PhysRevE.98.012134|pmid=30110832 |bibcode=2018PhRvE..98a2134L |s2cid=52017287}}

0.2854, 0.290,{{Cite journal|last1=Li |first1=Mingqi|last2=Chen|first2=Huisu|last3=Lin|first3=Jianjun|date=January 2020|title=Efficient measurement of the percolation threshold for random systems of congruent overlapping ovoids|journal=Powder Technology|volume=360|pages=598–607|doi=10.1016/j.powtec.2019.10.044|s2cid=208693526|issn=0032-5910}} 0.290,{{Cite journal|last1=Li |first1=Mingqi|last2=Chen|first2=Huisu|last3=Lin|first3=Jianjun |date=April 2020|title=Numerical study for the percolation threshold and transport properties of porous composites comprising non-centrosymmetrical superovoidal pores |journal=Computer Methods in Applied Mechanics and Engineering|volume=361|pages=112815|doi=10.1016/j.cma.2019.112815|bibcode=2020CMAME.361k2815L |s2cid=213152892|issn=0045-7825}} 0.2895693(26){{Cite journal

|last=Brzeski |first=Piotr

|last2=Kondrat |first2=Grzegorz

|date=2022

|title=Percolation of hyperspheres in dimensions 3 to 5: from discrete to continuous |url=https://iopscience.iop.org/article/10.1088/1742-5468/ac6519

|journal=Journal of Statistical Mechanics: Theory and Experiment

|volume=2022 |issue=5 |pages=053202 |doi=10.1088/1742-5468/ac6519 |issn=1742-5468}}

| 0.3418(7), 0.3438(13), 0.341889(3), 0.3360, 0.34189(2) [corrected], 0.341935(8), 0.335,

Oblate ellipsoids with major radius r and aspect ratio {{frac|4|3}}

| 0.2831{{cite journal

| last = Garboczi

| first = E. J.

|author2=K. A. Snyder |author3=J. F. Douglas

| title = Geometrical percolation threshold of overlapping ellipsoids

| journal = Physical Review E

| volume = 52

| issue = 1

| year = 1995

| pages = 819–827 |bibcode = 1995PhRvE..52..819G |doi = 10.1103/PhysRevE.52.819 | pmid = 9963485

| url = https://zenodo.org/record/1233793

}}

| 0.3328

Prolate ellipsoids with minor radius r and aspect ratio {{frac|3|2}}

| 0.2757, 0.2795, 0.2763

| 0.3278

Oblate ellipsoids with major radius r and aspect ratio 2

| 0.2537, 0.2629, 0.254

| 0.3050

Prolate ellipsoids with minor radius r and aspect ratio 2

| 0.2537, 0.2618, 0.25(2),{{cite journal

| last = Yi

| first = Y.-B.

|author2=A. M. Sastry

| title = Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution

| journal = Proc. R. Soc. Lond. A

| volume =460

| issue =2048

| year = 2004

| pages =2353–2380

| doi = 10.1098/rspa.2004.1279 |bibcode = 2004RSPSA.460.2353Y | s2cid = 2475482

}}

0.2507

| 0.3035, 0.29(3)

Oblate ellipsoids with major radius r and aspect ratio 3

| 0.2289

| 0.2599

Prolate ellipsoids with minor radius r and aspect ratio 3

| 0.2033, 0.2244, 0.20(2)

| 0.2541, 0.22(3)

Oblate ellipsoids with major radius r and aspect ratio 4

| 0.2003

| 0.2235

Prolate ellipsoids with minor radius r and aspect ratio 4

| 0.1901, 0.16(2)

| 0.2108, 0.17(3)

Oblate ellipsoids with major radius r and aspect ratio 5

| 0.1757

| 0.1932

Prolate ellipsoids with minor radius r and aspect ratio 5

| 0.1627, 0.13(2)

| 0.1776, 0.15(2)

Oblate ellipsoids with major radius r and aspect ratio 10

| 0.0895, 0.1058

| 0.1118

Prolate ellipsoids with minor radius r and aspect ratio 10

| 0.0724, 0.08703, 0.07(2)

| 0.09105, 0.07(2)

Oblate ellipsoids with major radius r and aspect ratio 100

| 0.01248

| 0.01256

Prolate ellipsoids with minor radius r and aspect ratio 100

| 0.006949

| 0.006973

Oblate ellipsoids with major radius r and aspect ratio 1000

| 0.001275

| 0.001276

Oblate ellipsoids with major radius r and aspect ratio 2000

| 0.000637

Spherocylinders with H/D = 1

| 0.2439(2)

Spherocylinders with H/D = 4

| 0.1345(1)

Spherocylinders with H/D = 10

| 0.06418(20)

Spherocylinders with H/D = 50

| 0.01440(8)

Spherocylinders with H/D = 100

| 0.007156(50)

Spherocylinders with H/D = 200

| 0.003724(90)

Aligned cylinders

| 0.2819(2)

| 0.3312(1){{cite journal

| last = Hyytiä

| first = E.

| author2 = J. Virtamo |author3=P. Lassila |author4=J. Ott

| title = Continuum percolation threshold for permeable aligned cylinders and opportunistic networking

| journal = IEEE Communications Letters

| volume = 16

| issue = 7

| year = 2012

| pages = 1064–1067

| doi = 10.1109/LCOMM.2012.051512.120497 | s2cid = 1056865

| url = https://zenodo.org/record/1223952

}}

Aligned cubes of side

\ell = 2 a

| 0.2773(2) 0.27727(2), 0.27730261(79)

| 0.3247(3), 0.3248(3), 0.32476(4) 0.324766(1)

Randomly oriented icosahedra

| 0.3030(5)

Randomly oriented dodecahedra

| 0.2949(5)

Randomly oriented octahedra

| 0.2514(6)

Randomly oriented cubes of side

\ell = 2 a

| 0.2168(2) 0.2174,

| 0.2444(3), 0.2443(5){{cite journal

| last = Torquato

| first = S.

|author2=Y. Jiao

| title = Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles

| journal = Physical Review E

| volume = 87

| issue = 2

| year = 2012

| arxiv = 1210.0134

| pages = 022111

| doi = 10.1103/PhysRevE.87.022111| pmid = 23496464

|bibcode = 2013PhRvE..87b2111T | s2cid = 11417012

}}

Randomly oriented tetrahedra

| 0.1701(7)

Randomly oriented disks of radius r (in 3D)

| 0.9614(5){{cite journal

| last = Yi

| first = Y. B.

|author2=E. Tawerghi

| title = Geometric percolation thresholds of interpenetrating plates in three-dimensional space

| journal = Physical Review E

| volume = 79

| issue = 4

| year = 2009

| pages = 041134

| doi = 10.1103/PhysRevE.79.041134 | pmid = 19518200

|bibcode = 2009PhRvE..79d1134Y }}

Randomly oriented square plates of side

\sqrt{\pi} r

| 0.8647(6)

Randomly oriented triangular plates of side

\sqrt{2 \pi} /3^{1/4} r

| 0.7295(6)

Jammed spheres (average z = 6)

| 0.183(3), 0.1990,

{{cite journal

| last = Ziff

| first = R. M.

|author2=Salvatore Torquato

| title = Percolation of disordered jammed sphere packings

| journal = Journal of Physics A: Mathematical and Theoretical

| volume =50

| issue = 8

| year = 2016

| pages = 085001

| arxiv = 1611.00279| bibcode =2017JPhA...50h5001Z| doi =10.1088/1751-8121/aa5664| s2cid = 53003822

}}

see also contact network of jammed spheres below.

| 0.59(1) (volume fraction of all spheres)

$\eta_c = (4/3) \pi r^3 N / L^3$ is the total volume (for spheres), where N is the number of objects and L is the system size.

$\phi_c = 1 - e^{-\eta_c}$ is the critical volume fraction, valid for overlapping randomly placed objects.

For disks and plates, these are effective volumes and volume fractions.

For void ("Swiss-Cheese" model), $\phi_c = e^{-\eta_c}$ is the critical void fraction.

For more results on void percolation around ellipsoids and elliptical plates, see.

For more ellipsoid percolation values see.

For spherocylinders, H/D is the ratio of the height to the diameter of the cylinder, which is then capped by hemispheres. Additional values are given in.

For superballs, m is the deformation parameter, the percolation values are given in.,{{cite journal |last1=Lin |first1=Jianjun |last2=Chen |first2=Huisu |title=Continuum percolation of porous media via random packing of overlapping cube-like particles |journal=Theoretical & Applied Mechanics Letters |date=2018 |volume=8 |issue=5 |pages=299–303 |doi=10.1016/j.taml.2018.05.007|doi-access=free }}{{cite journal |last1=Lin |first1=Jianjun |last2=Chen |first2=Huisu |title=Effect of particle morphologies on the percolation of particulate porous media: A study of superballs |journal=Powder Technology |date=2018 |volume=335 |pages=388–400 |doi=10.1016/j.powtec.2018.05.015|s2cid=103471554 }} In addition, the thresholds of concave-shaped superballs are also determined in

For cuboid-like particles (superellipsoids), m is the deformation parameter, more percolation values are given in.

= Void percolation in 3D =

Void percolation refers to percolation in the space around overlapping objects. Here $\phi_c$ refers to the fraction of the space occupied by the voids (not of the particles) at the critical point, and is related to $\eta_c$ by

$\phi_c = e^{-\eta_c}$ . $\eta_c$ is defined as in the continuum percolation section above.

class="wikitable"
System ! Φ_c ! η_c
Voids around disks of radius r \| \| 22.86(2)
Voids around randomly oriented tetrahedra \| 0.0605(6) \|
Voids around oblate ellipsoids of major radius r and aspect ratio 32 \| 0.5308(7) \| 0.6333
Voids around oblate ellipsoids of major radius r and aspect ratio 16 \| 0.3248(5) \| 1.125
Voids around oblate ellipsoids of major radius r and aspect ratio 10 \| \| 1.542(1)
Voids around oblate ellipsoids of major radius r and aspect ratio 8 \| 0.1615(4) \| 1.823
Voids around oblate ellipsoids of major radius r and aspect ratio 4 \| 0.0711(2) \| 2.643, 2.618(5)
Voids around oblate ellipsoids of major radius r and aspect ratio 2 \| \| 3.239(4)
Voids around prolate ellipsoids of aspect ratio 8 \| 0.0415(7) \|
Voids around prolate ellipsoids of aspect ratio 6 \| 0.0397(7) \|
Voids around prolate ellipsoids of aspect ratio 4 \| 0.0376(7) \|
Voids around prolate ellipsoids of aspect ratio 3 \| 0.03503(50) \|
Voids around prolate ellipsoids of aspect ratio 2 \| 0.0323(5) \|
Voids around aligned square prisms of aspect ratio 2 \| 0.0379(5) \|
Voids around randomly oriented square prisms of aspect ratio 20 \| 0.0534(4) \|
Voids around randomly oriented square prisms of aspect ratio 15 \| 0.0535(4) \|
Voids around randomly oriented square prisms of aspect ratio 10 \| 0.0524(5) \|
Voids around randomly oriented square prisms of aspect ratio 8 \| 0.0523(6) \|
Voids around randomly oriented square prisms of aspect ratio 7 \| 0.0519(3) \|
Voids around randomly oriented square prisms of aspect ratio 6 \| 0.0519(5) \|
Voids around randomly oriented square prisms of aspect ratio 5 \| 0.0515(7) \|
Voids around randomly oriented square prisms of aspect ratio 4 \| 0.0505(7) \|
Voids around randomly oriented square prisms of aspect ratio 3 \| 0.0485(11) \|
Voids around randomly oriented square prisms of aspect ratio 5/2 \| 0.0483(8) \|
Voids around randomly oriented square prisms of aspect ratio 2 \| 0.0465(7) \|
Voids around randomly oriented square prisms of aspect ratio 3/2 \| 0.0461(14) \|
Voids around hemispheres \| 0.0455(6) \|
Voids around aligned tetrahedra \| 0.0605(6) \|
Voids around randomly oriented tetrahedra \| 0.0605(6) \|
Voids around aligned cubes \| 0.036(1), 0.0381(3) \|
Voids around randomly oriented cubes \| 0.0452(6), 0.0449(5) \|
Voids around aligned octahedra \| 0.0407(3) \|
Voids around randomly oriented octahedra \| 0.0398(5) \|
Voids around aligned dodecahedra \| 0.0356(3) \|
Voids around randomly oriented dodecahedra \| 0.0360(3) \|
Voids around aligned icosahedra \| 0.0346(3) \|
Voids around randomly oriented icosahedra \| 0.0336(7) \|
Voids around spheres \| 0.034(7),{{cite journal \| last = Kertesz \| first = Janos \| journal = Journal de Physique Lettres \| title = Percolation of holes between overlapping spheres: Monte Carlo calculation of the critical volume fraction \| volume = 42 \| issue = 17 \| year = 1981 \| pages = L393– L395 \| doi = 10.1051/jphyslet:019810042017039300\| s2cid = 122115573 \| url = https://hal.archives-ouvertes.fr/jpa-00231955/file/ajp-jphyslet_1981_42_17_393_0.pdf }} 0.032(4),{{cite journal \| last = Elam \| first = W. T. \| author2 = A. R. Kerstein \| author3 = J. J. Rehr \| journal = Physical Review Letters \| title = Critical properties of the void percolation problem for spheres \| volume = 52 \| issue = 7 \| year = 1984 \| pages = 1516–1519 \| doi = 10.1103/PhysRevLett.52.1516\| bibcode = 1984PhRvL..52.1516E }} 0.030(2),{{cite journal \| last = van der Marck \| first = Steven C. \| title = Network approach to void percolation in a pack of unequal spheres \| journal = Physical Review Letters \| volume = 77 \| issue = 9 \| year = 1996 \| pages = 1785–1788 \| doi = 10.1103/PhysRevLett.77.1785 \| pmid = 10063171 \| bibcode=1996PhRvL..77.1785V}} 0.0301(3),{{cite journal \| last = Rintoul \| first = M. D. \| title = Precise determination of the void percolation threshold for two distributions of overlapping spheres \| journal = Physical Review E \| volume = 62 \| issue = 6 \| year = 2000 \| pages = 68–72 \| doi = 10.1103/PhysRevE.62.68 \| pmid = 11088435 \| bibcode = 2000PhRvE..62...68R \| url = https://digital.library.unt.edu/ark:/67531/metadc703993/ }} 0.0294,{{cite journal \| last = Yi \| first = Y. B. \| title = Void percolation and conduction of overlapping ellipsoids \| journal = Physical Review E \| volume = 74 \| issue = 3 \| year = 2006 \| pages = 031112 \| doi = 10.1103/PhysRevE.74.031112\| pmid = 17025599 \|bibcode = 2006PhRvE..74c1112Y }} 0.0300(3),{{cite journal \| last = Höfling \| first = F. \|author2=T. Munk \|author3=E. Frey \|author4=T. Franosch \| title = Critical dynamics of ballistic and Brownian particles in a heterogeneous environment \| journal = J. Chem. Phys. \| volume = 128 \| issue = 16 \| year = 2008 \| pages = 164517 \| doi = 10.1063/1.2901170\| pmid = 18447469 \|bibcode = 2008JChPh.128p4517H \|arxiv = 0712.2313 \| s2cid = 25509814 }} 0.0317(4),{{cite journal \| last = Priour, Jr. \| first = D.J. \| title = Percolation through voids around overlapping spheres: A dynamically based finite-size scaling analysis \| journal = Physical Review E \| volume = 89 \| issue = 1 \| year = 2014 \| pages = 012148 \| doi = 10.1103/PhysRevE.89.012148 \| pmid = 24580213 \|bibcode = 2014PhRvE..89a2148P \| arxiv = 1208.0328\| s2cid = 20349307 }} 0.0308(5){{cite arXiv \| last = Priour, Jr. \| first = D. J. \| author2 = N. J. McGuigan \| title = Percolation through voids around randomly oriented faceted inclusions \| year = 2017 \| eprint = 1712.10241 \| class = cond-mat.stat-mech }} 0.0301(1), 0.0301(1){{cite journal \| last = Priour, Jr. \| first = D. J. \|author2 = N. J. McGuigan \| title = Percolation through voids around randomly oriented polyhedra and axially symmetric grains \| year = 2018 \| journal = Physical Review Letters \| volume = 121 \| issue = 22 \| pages = 225701 \| doi = 10.1103/PhysRevLett.121.225701 \| pmid = 30547614 \| bibcode = 2018PhRvL.121v5701P \| arxiv = 1801.09970 \| s2cid = 119185480 }} \| 3.506(8), 3.515(6),{{cite journal \| last = Yi \| first = Y. B. \|author2=K. Esmail \| title = Computational measurement of void percolation thresholds of oblate particles and thin plate composites \| journal = J. Appl. Phys. \| volume = 111 \| issue = 12 \| year = 2012 \| pages = 124903–124903–6 \| doi = 10.1063/1.4730333\|bibcode = 2012JAP...111l4903Y }} 3.510(2){{cite journal \| last = Charbonneau \| first = Benoit \| author2 = Patrick Charbonneau \| author3 = Yi Hu \| author4 = Zhen Yang \| title = High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas \| year = 2021 \| journal = Physical Review E \| issue = 2 \| volume = 104 \| pages = 024137 \| doi = 10.1103/PhysRevE.104.024137 \| pmid = 34525662 \| arxiv = 2105.04711\| bibcode = 2021PhRvE.104b4137C \| s2cid = 234357912 }}

class="wikitable"

System

! Φ_c

! η_c

Voids around disks of radius r

| 22.86(2)

Voids around randomly oriented tetrahedra

| 0.0605(6)

Voids around oblate ellipsoids of major radius r and aspect ratio 32

| 0.5308(7)

| 0.6333

Voids around oblate ellipsoids of major radius r and aspect ratio 16

| 0.3248(5)

| 1.125

Voids around oblate ellipsoids of major radius r and aspect ratio 10

| 1.542(1)

Voids around oblate ellipsoids of major radius r and aspect ratio 8

| 0.1615(4)

| 1.823

Voids around oblate ellipsoids of major radius r and aspect ratio 4

| 0.0711(2)

| 2.643, 2.618(5)

Voids around oblate ellipsoids of major radius r and aspect ratio 2

| 3.239(4)

Voids around prolate ellipsoids of aspect ratio 8

| 0.0415(7)

Voids around prolate ellipsoids of aspect ratio 6

| 0.0397(7)

Voids around prolate ellipsoids of aspect ratio 4

| 0.0376(7)

Voids around prolate ellipsoids of aspect ratio 3

| 0.03503(50)

Voids around prolate ellipsoids of aspect ratio 2

| 0.0323(5)

Voids around aligned square prisms of aspect ratio 2

| 0.0379(5)

Voids around randomly oriented square prisms of aspect ratio 20

| 0.0534(4)

Voids around randomly oriented square prisms of aspect ratio 15

| 0.0535(4)

Voids around randomly oriented square prisms of aspect ratio 10

| 0.0524(5)

Voids around randomly oriented square prisms of aspect ratio 8

| 0.0523(6)

Voids around randomly oriented square prisms of aspect ratio 7

| 0.0519(3)

Voids around randomly oriented square prisms of aspect ratio 6

| 0.0519(5)

Voids around randomly oriented square prisms of aspect ratio 5

| 0.0515(7)

Voids around randomly oriented square prisms of aspect ratio 4

| 0.0505(7)

Voids around randomly oriented square prisms of aspect ratio 3

| 0.0485(11)

Voids around randomly oriented square prisms of aspect ratio 5/2

| 0.0483(8)

Voids around randomly oriented square prisms of aspect ratio 2

| 0.0465(7)

Voids around randomly oriented square prisms of aspect ratio 3/2

| 0.0461(14)

Voids around hemispheres

| 0.0455(6)

Voids around aligned tetrahedra

| 0.0605(6)

Voids around randomly oriented tetrahedra

| 0.0605(6)

Voids around aligned cubes

| 0.036(1), 0.0381(3)

Voids around randomly oriented cubes

| 0.0452(6), 0.0449(5)

Voids around aligned octahedra

| 0.0407(3)

Voids around randomly oriented octahedra

| 0.0398(5)

Voids around aligned dodecahedra

| 0.0356(3)

Voids around randomly oriented dodecahedra

| 0.0360(3)

Voids around aligned icosahedra

| 0.0346(3)

Voids around randomly oriented icosahedra

| 0.0336(7)

Voids around spheres

| 0.034(7),{{cite journal

| last = Kertesz

| first = Janos

| journal = Journal de Physique Lettres

| title = Percolation of holes between overlapping spheres: Monte Carlo calculation of the critical volume fraction

| volume = 42

| issue = 17

| year = 1981

| pages = L393– L395

| doi = 10.1051/jphyslet:019810042017039300| s2cid = 122115573

| url = https://hal.archives-ouvertes.fr/jpa-00231955/file/ajp-jphyslet_1981_42_17_393_0.pdf

}} 0.032(4),{{cite journal

| last = Elam

| first = W. T.

| author2 = A. R. Kerstein

| author3 = J. J. Rehr

| journal = Physical Review Letters

| title = Critical properties of the void percolation problem for spheres

| volume = 52

| issue = 7

| year = 1984

| pages = 1516–1519

| doi = 10.1103/PhysRevLett.52.1516| bibcode = 1984PhRvL..52.1516E

}} 0.030(2),{{cite journal

| last = van der Marck

| first = Steven C.

| title = Network approach to void percolation in a pack of unequal spheres

| journal = Physical Review Letters

| volume = 77

| issue = 9

| year = 1996

| pages = 1785–1788

| doi = 10.1103/PhysRevLett.77.1785

| pmid = 10063171

| bibcode=1996PhRvL..77.1785V}}

0.0301(3),{{cite journal

| last = Rintoul

| first = M. D.

| title = Precise determination of the void percolation threshold for two distributions of overlapping spheres

| journal = Physical Review E

| volume = 62

| issue = 6

| year = 2000

| pages = 68–72

| doi = 10.1103/PhysRevE.62.68

| pmid = 11088435

| bibcode = 2000PhRvE..62...68R

| url = https://digital.library.unt.edu/ark:/67531/metadc703993/

}}

0.0294,{{cite journal

| last = Yi

| first = Y. B.

| title = Void percolation and conduction of overlapping ellipsoids

| journal = Physical Review E

| volume = 74

| issue = 3

| year = 2006

| pages = 031112

| doi = 10.1103/PhysRevE.74.031112| pmid = 17025599

|bibcode = 2006PhRvE..74c1112Y }}

0.0300(3),{{cite journal

| last = Höfling

| first = F.

|author2=T. Munk |author3=E. Frey |author4=T. Franosch

| title = Critical dynamics of ballistic and Brownian particles in a heterogeneous environment

| journal = J. Chem. Phys.

| volume = 128

| issue = 16

| year = 2008

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| doi = 10.1063/1.2901170| pmid = 18447469

|bibcode = 2008JChPh.128p4517H |arxiv = 0712.2313 | s2cid = 25509814

}}

0.0317(4),{{cite journal

| last = Priour, Jr.

| first = D.J.

| title = Percolation through voids around overlapping spheres: A dynamically based finite-size scaling analysis

| journal = Physical Review E

| volume = 89

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| year = 2014

| pages = 012148

| doi = 10.1103/PhysRevE.89.012148

| pmid = 24580213

|bibcode = 2014PhRvE..89a2148P | arxiv = 1208.0328| s2cid = 20349307

}}

0.0308(5){{cite arXiv

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| first = D. J.

| author2 = N. J. McGuigan

| title = Percolation through voids around randomly oriented faceted inclusions

| year = 2017

| eprint = 1712.10241

| class = cond-mat.stat-mech

}}

0.0301(1), 0.0301(1){{cite journal

| last = Priour, Jr.

| first = D. J.

|author2 = N. J. McGuigan

| title = Percolation through voids around randomly oriented polyhedra and axially symmetric grains

| year = 2018

| journal = Physical Review Letters

| volume = 121

| issue = 22

| pages = 225701

| doi = 10.1103/PhysRevLett.121.225701

| pmid = 30547614

| bibcode = 2018PhRvL.121v5701P

| arxiv = 1801.09970

| s2cid = 119185480

}}

| 3.506(8), 3.515(6),{{cite journal

| last = Yi

| first = Y. B.

|author2=K. Esmail

| title = Computational measurement of void percolation thresholds of oblate particles and thin plate composites

| journal = J. Appl. Phys.

| volume = 111

| issue = 12

| year = 2012

| pages = 124903–124903–6

| doi = 10.1063/1.4730333|bibcode = 2012JAP...111l4903Y }}

3.510(2){{cite journal

| last = Charbonneau

| first = Benoit

| author2 = Patrick Charbonneau

| author3 = Yi Hu

| author4 = Zhen Yang

| title = High-dimensional percolation criticality and hints of mean-field-like caging of the random Lorentz gas

| year = 2021

| journal = Physical Review E

| issue = 2

| volume = 104

| pages = 024137

| doi = 10.1103/PhysRevE.104.024137

| pmid = 34525662

| arxiv = 2105.04711| bibcode = 2021PhRvE.104b4137C

| s2cid = 234357912

}}

= Thresholds on 3D random and quasi-lattices =

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Bond percolation threshold
Contact network of packed spheres \| \| 6 \| 0.310(5), {{cite journal \| last = Powell \| first = M. J. \| title = Site percolation in randomly packed spheres \| journal = Physical Review B \| volume = 20 \| issue = 10 \| year = 1979 \| pages =4194–4198 \| doi=10.1103/PhysRevB.20.4194\|bibcode = 1979PhRvB..20.4194P }} 0.287(50), {{cite journal \| last = Clerc \| first = J. P. \| author2 = G. Giraud \| author3 = S. Alexander \| author4 = E. Guyon \| title = Conductivity of a mixture of conducting and insulating grains: Dimensionality effects \| journal = Physical Review B \| volume = 22 \| issue = 5 \| year = 1979 \| pages =2489–2494 \| doi=10.1103/PhysRevB.22.2489}} 0.3116(3), \|
Random-plane tessellation, dual \| \| 6 \| 0.290(7) {{Cite journal \| author1=C. Larmier \|author2=E. Dumonteil \|author3=F. Malvagi \|author4=A. Mazzolo \|author5=A. Zoia \| title = Finite-size effects and percolation properties of Poisson geometries \| journal = Physical Review E \| volume = 94 \| issue =1 \| pages = 012130 \| arxiv=1605.04550 \| year = 2016 \| doi = 10.1103/PhysRevE.94.012130 \| pmid = 27575099 \| bibcode = 2016PhRvE..94a2130L\| s2cid = 19361619 }} \|
Icosahedral Penrose \| \| 6 \| 0.285{{cite journal \| last = Zakalyukin \| first = R. M. \|author2=V. A. Chizhikov \| title = Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method \| journal = Crystallography Reports \| volume = 50 \| issue = 6 \| year = 2005 \| pages = 938–948 \| doi =10.1134/1.2132400 \|bibcode = 2005CryRp..50..938Z \| s2cid = 94290876 }} \| 0.225
Penrose w/2 diagonals \| \| 6.764 \| 0.271 \| 0.207
Penrose w/8 diagonals \| \| 12.764 \| 0.188 \| 0.111
Voronoi network \| \| 15.54 \| 0.1453(20) {{cite journal \| last = Jerauld \| first = G. R. \|author2=L. E. Scriven\|author3=H. T. Davis \| title =Percolation and conduction on the 3D Voronoi and regular networks: a second case study in topological disorder \| journal = J. Phys. C: Solid State Phys. \| volume = 17 \| issue = 19 \| year = 1984 \| pages =3429–3439 \| doi=10.1088/0022-3719/17/19/017\|bibcode = 1984JPhC...17.3429J }} \| 0.0822(50)

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Bond percolation threshold

Contact network of packed spheres

| 6

| 0.310(5),

{{cite journal

| last = Powell

| first = M. J.

| title = Site percolation in randomly packed spheres

| journal = Physical Review B

| volume = 20

| issue = 10

| year = 1979

| pages =4194–4198

| doi=10.1103/PhysRevB.20.4194|bibcode = 1979PhRvB..20.4194P }}

0.287(50),

{{cite journal

| last = Clerc

| first = J. P.

| author2 = G. Giraud

| author3 = S. Alexander

| author4 = E. Guyon

| title = Conductivity of a mixture of conducting and insulating grains: Dimensionality effects

| journal = Physical Review B

| volume = 22

| issue = 5

| year = 1979

| pages =2489–2494

| doi=10.1103/PhysRevB.22.2489}}

0.3116(3),

Random-plane tessellation, dual

| 6

| 0.290(7)

{{Cite journal

| title = Finite-size effects and percolation properties of Poisson geometries

| journal = Physical Review E

| volume = 94

| issue =1

| pages = 012130

| arxiv=1605.04550

| year = 2016

| doi = 10.1103/PhysRevE.94.012130

| pmid = 27575099

| bibcode = 2016PhRvE..94a2130L| s2cid = 19361619

}}

Icosahedral Penrose

| 6

| 0.285{{cite journal

| last = Zakalyukin

| first = R. M.

|author2=V. A. Chizhikov

| title = Calculations of the Percolation Thresholds of a Three-Dimensional (Icosahedral) Penrose Tiling by the Cubic Approximant Method

| journal = Crystallography Reports

| volume = 50

| issue = 6

| year = 2005

| pages = 938–948

| doi =10.1134/1.2132400 |bibcode = 2005CryRp..50..938Z | s2cid = 94290876

}}

| 0.225

Penrose w/2 diagonals

| 6.764

| 0.271

| 0.207

Penrose w/8 diagonals

| 12.764

| 0.188

| 0.111

Voronoi network

| 15.54

| 0.1453(20)

{{cite journal

| last = Jerauld

| first = G. R.

|author2=L. E. Scriven|author3=H. T. Davis

| title =Percolation and conduction on the 3D Voronoi and regular networks: a second case study in topological disorder

| journal = J. Phys. C: Solid State Phys.

| volume = 17

| issue = 19

| year = 1984

| pages =3429–3439

| doi=10.1088/0022-3719/17/19/017|bibcode = 1984JPhC...17.3429J }}

| 0.0822(50)

= Thresholds for other 3D models =

class="wikitable"
Lattice ! z ! $\overline z$ ! Site percolation threshold ! Critical coverage fraction $\phi_c$ ! Bond percolation threshold
Drilling percolation, simple cubic lattice* \| 6 \| 6 \| 0.6345(3), {{cite journal \| last = Kantor \| first = Yacov \| title = Three-dimensional percolation with removed lines of sites \| journal = Physical Review B \| volume = 33 \| issue = 5 \| year = 1986 \| pages = 3522–3525 \| doi = 10.1103/PhysRevB.33.3522\| pmid = 9938740 \| bibcode = 1986PhRvB..33.3522K }} 0.6339(5), {{cite journal \| last = Schrenk \| first = K. J. \| author2 = M. R. Hilário \| author3 = V. Sidoravicius \| author4 = N. A. M. Araújo \| author5 = H. J. Herrmann \| author6 = M. Thielmann \| author7 = A. Teixeira \| title = Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube? \| journal = Physical Review Letters \| volume = 116 \| issue = 5 \| year = 2016 \| pages = 055701 \| arxiv =1601.03534 \| doi= 10.1103/PhysRevLett.116.055701 \| pmid = 26894717 \| bibcode=2016PhRvL.116e5701S\| s2cid = 3145131 }} 0.633965(15) {{cite journal \| last = Grassberger \| first = P. \| title = Some remarks on drilling percolation \| journal = Physical Review E \| volume = 95 \| issue = 1 \| year = 2017 \| pages =010103 \| arxiv =1611.07939 \| doi=10.1103/PhysRevE.95.010103\| pmid = 28208497 \| s2cid = 12476714 \| url = http://juser.fz-juelich.de/search?p=id:%22FZJ-2017-07804%22 }} \| 0.25480 \|
Drill in z direction on cubic lattice, remove single sites \| 6 \| 6 \| 0.592746 (columns), 0.4695(10) (sites) \| 0.2784 \|
Random tube model, simple cubic lattice^† \| \| \| \| \| 0.231456(6) {{cite journal \| last = Szczygieł \| first = Bartłomiej \| author2 = Kamil Kwiatkowski \| author3 = Maciej Lewenstein \| author4 = Gerald John Lapeyre, Jr. \| author5 = Jan Wehr \| title = Percolation thresholds for discrete-continuous models with nonuniform probabilities of bond formation \| journal = Physical Review E \| volume = 93 \| issue = 2 \| year = 2016 \| pages = 022127 \| arxiv = 1509.07401 \| doi=10.1103/PhysRevE.93.022127 \| pmid = 26986308 \| bibcode = 2016PhRvE..93b2127S \| s2cid = 18110437 }}
Pac-Man percolation, simple cubic lattice \| \| \| \| \| 0.139(6) {{cite journal \| last = Abete \| first = T. \| author2 = A. de Candia \| author3 = D. Lairez \| author4 = A. Coniglio \| title = Percolation Model for Enzyme Gel Degradation \| journal = Physical Review Letters \| volume = 93 \| issue = \| year = 2004 \| pages = 228301 \| arxiv =cond-mat/0402551 \| doi=10.1103/PhysRevLett.93.228301 }}

class="wikitable"

Lattice

! z

! $\overline z$

! Site percolation threshold

! Critical coverage fraction $\phi_c$

! Bond percolation threshold

Drilling percolation, simple cubic lattice*

| 6

| 0.6345(3),

{{cite journal

| last = Kantor

| first = Yacov

| title = Three-dimensional percolation with removed lines of sites

| journal = Physical Review B

| volume = 33

| issue = 5

| year = 1986

| pages = 3522–3525

| doi = 10.1103/PhysRevB.33.3522| pmid = 9938740

| bibcode = 1986PhRvB..33.3522K

}}

0.6339(5),

{{cite journal

| last = Schrenk

| first = K. J.

| author2 = M. R. Hilário

| author3 = V. Sidoravicius

| author4 = N. A. M. Araújo

| author5 = H. J. Herrmann

| author6 = M. Thielmann

| author7 = A. Teixeira

| title = Critical Fragmentation Properties of Random Drilling: How Many Holes Need to Be Drilled to Collapse a Wooden Cube?

| journal = Physical Review Letters

| volume = 116

| issue = 5

| year = 2016

| pages = 055701

| arxiv =1601.03534 | doi= 10.1103/PhysRevLett.116.055701

| pmid = 26894717

| bibcode=2016PhRvL.116e5701S| s2cid = 3145131

}}

0.633965(15)

{{cite journal

| last = Grassberger

| first = P.

| title = Some remarks on drilling percolation

| journal = Physical Review E

| volume = 95

| issue = 1

| year = 2017

| pages =010103

| arxiv =1611.07939 | doi=10.1103/PhysRevE.95.010103| pmid = 28208497

| s2cid = 12476714

| url = http://juser.fz-juelich.de/search?p=id:%22FZJ-2017-07804%22

}}

| 0.25480

Drill in z direction on cubic lattice, remove single sites

| 6

| 0.592746 (columns), 0.4695(10) (sites)

| 0.2784

Random tube model, simple cubic lattice^†

| 0.231456(6)

{{cite journal

| last = Szczygieł

| first = Bartłomiej

| author2 = Kamil Kwiatkowski

| author3 = Maciej Lewenstein

| author4 = Gerald John Lapeyre, Jr.

| author5 = Jan Wehr

| title = Percolation thresholds for discrete-continuous models with nonuniform probabilities of bond formation

| journal = Physical Review E

| volume = 93

| issue = 2

| year = 2016

| pages = 022127

| arxiv = 1509.07401

| doi=10.1103/PhysRevE.93.022127

| pmid = 26986308

| bibcode = 2016PhRvE..93b2127S

| s2cid = 18110437

}}

Pac-Man percolation, simple cubic lattice

| 0.139(6)

{{cite journal

| last = Abete

| first = T.

| author2 = A. de Candia

| author3 = D. Lairez

| author4 = A. Coniglio

| title = Percolation Model for Enzyme Gel Degradation

| journal = Physical Review Letters

| volume = 93

| issue =

| year = 2004

| pages = 228301

| arxiv =cond-mat/0402551

| doi=10.1103/PhysRevLett.93.228301

}}

$^*$ In drilling percolation, the site threshold $p_c$ represents the fraction of columns in each direction that have not been removed, and $\phi_c=p_c^3$ . For the 1d drilling, we have $\phi_c = p_c$ (columns) $p_c$ (sites).

^† In tube percolation, the bond threshold represents the value of the parameter $\mu$ such that the probability of putting a bond between neighboring vertical tube segments is $1-e^{-\mu h_i}$ , where $h_i$ is the overlap height of two adjacent tube segments.

Thresholds in different dimensional spaces

= Continuum models in higher dimensions =

class="wikitable"
d ! System ! Φ_c ! η_c
4 \| Overlapping hyperspheres \| 0.1223(4) \| 0.1300(13), 0.1304(5), 0.1210268(19)
4 \| Aligned hypercubes \| 0.1132(5), 0.1132348(17) \| 0.1201(6)
4 \| Voids around hyperspheres \| 0.00211(2) \| 6.161(10) 6.248(2),
5 \| Overlapping hyperspheres \| \| 0.0544(6), 0.05443(7), 0.0522524(69)
5 \| Aligned hypercubes \| 0.04900(7), 0.0481621(13) \| 0.05024(7)
5 \| Voids around hyperspheres \| 1.26(6)x10⁻⁴ \| 8.98(4), 9.170(8)
6 \| Overlapping hyperspheres \| \| 0.02391(31), 0.02339(5)
6 \| Aligned hypercubes \| 0.02082(8), 0.0213479(10) \| 0.02104(8)
6 \| Voids around hyperspheres \| 8.0(6)x10⁻⁶ \| 11.74(8), 12.24(2),
7 \| Overlapping hyperspheres \| \| 0.01102(16), 0.01051(3)
7 \| Aligned hypercubes \| 0.00999(5), 0.0097754(31) \| 0.01004(5)
7 \| Voids around hyperspheres \| \| 15.46(5)
8 \| Overlapping hyperspheres \| \| 0.00516(8), 0.004904(6)
8 \| Aligned hypercubes \| \| 0.004498(5)
8 \| Voids around hyperspheres \| \| 18.64(8)
9 \| Overlapping hyperspheres \| \| 0.002353(4)
9 \| Aligned hypercubes \| \| 0.002166(4)
9 \| Voids around hyperspheres \| \| 22.1(4)
10 \| Overlapping hyperspheres \| \| 0.001138(3)
10 \| Aligned hypercubes \| \| 0.001058(4)
11 \| Overlapping hyperspheres \| \| 0.0005530(3)
11 \| Aligned hypercubes \| \| 0.0005160(3)

class="wikitable"

! System

! Φ_c

! η_c

| Overlapping hyperspheres

| 0.1223(4)

| 0.1300(13), 0.1304(5), 0.1210268(19)

| Aligned hypercubes

| 0.1132(5), 0.1132348(17)

| 0.1201(6)

| Voids around hyperspheres

| 0.00211(2)

| 6.161(10) 6.248(2),

| Overlapping hyperspheres

| 0.0544(6), 0.05443(7), 0.0522524(69)

| Aligned hypercubes

| 0.04900(7), 0.0481621(13)

| 0.05024(7)

| Voids around hyperspheres

| 1.26(6)x10⁻⁴

| 8.98(4), 9.170(8)

| Overlapping hyperspheres

| 0.02391(31), 0.02339(5)

| Aligned hypercubes

| 0.02082(8), 0.0213479(10)

| 0.02104(8)

| Voids around hyperspheres

| 8.0(6)x10⁻⁶

| 11.74(8), 12.24(2),

| Overlapping hyperspheres

| 0.01102(16), 0.01051(3)

| Aligned hypercubes

| 0.00999(5), 0.0097754(31)

| 0.01004(5)

| Voids around hyperspheres

| 15.46(5)

| Overlapping hyperspheres

| 0.00516(8), 0.004904(6)

| Aligned hypercubes

| 0.004498(5)

| Voids around hyperspheres

| 18.64(8)

| Overlapping hyperspheres

| 0.002353(4)

| Aligned hypercubes

| 0.002166(4)

| Voids around hyperspheres

| 22.1(4)

| Overlapping hyperspheres

| 0.001138(3)

| Aligned hypercubes

| 0.001058(4)

| Overlapping hyperspheres

| 0.0005530(3)

| Aligned hypercubes

| 0.0005160(3)

$\eta_c = (\pi^{d/2}/ \Gamma[d/2 + 1]) r^d N / L^d.$

In 4d, $\eta_c = (1/2) \pi^2 r^4 N / L^4$ .

In 5d, $\eta_c = (8/15) \pi^2 r^5 N / L^5$ .

In 6d, $\eta_c = (1/6) \pi^3 r^6 N / L^6$ .

$\phi_c = 1 - e^{-\eta_c}$ is the critical volume fraction, valid for overlapping objects.

For void models, $\phi_c = e^{-\eta_c}$ is the critical void fraction, and $\eta_c$ is the total volume of the overlapping objects

=Thresholds on hypercubic lattices=

class="wikitable" border="1"
d ! z ! Site thresholds ! Bond thresholds
4 \| 8 \| 0.198(1){{cite journal \| last = Kirkpatrick \| first = Scott \| title = Percolation phenomena in higher dimensions: Approach to the mean-field limit \| journal = Physical Review Letters \| volume = 36 \| issue = 2 \| year = 1976 \| pages = 69–72 \| doi = 10.1103/PhysRevLett.36.69\| bibcode = 1976PhRvL..36...69K }} 0.197(6), 0.1968861(14),{{cite journal \| last = Grassberger \| first = Peter \| title = Critical percolation in high dimensions \| journal = Physical Review E \| volume = 67 \| issue = 3 \| year = 2003 \| pages = 4 \| doi = 10.1103/PhysRevE.67.036101 \| pmid = 12689126 \|arxiv = cond-mat/0202144 \|bibcode = 2003PhRvE..67c6101G \| s2cid = 43707822 }} 0.196889(3),{{cite journal \| last = Paul \| first = Gerald \|author2=Robert M. Ziff \|author3=H. Eugene Stanley \| title = Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions \| journal = Physical Review E \| volume = 64 \| issue = 2 \| year = 2001 \| pages = 8 \| doi = 10.1103/PhysRevE.64.026115 \| pmid = 11497659 \|arxiv = cond-mat/0101136 \|bibcode = 2001PhRvE..64b6115P \| s2cid = 18271196 }} 0.196901(5),{{cite journal \| last = Ballesteros \| first = H. G. \|author2=L. A. Fernández \|author3=V. Martín-Mayor \|author4=A. Muñoz Sudupe \|author5=G. Parisi \|author6=J. J. Ruiz-Lorenzo \| title = Measures of critical exponents in the four dimensional site percolation \| journal = Phys. Lett. B \| volume = 400 \| issue = 3–4 \| year = 1997 \| pages = 346–351 \| doi = 10.1016/S0370-2693(97)00337-7 \|arxiv = hep-lat/9612024 \|bibcode = 1997PhLB..400..346B \| s2cid = 10242417 }} 0.19680(23), {{cite journal \| last = Kotwica \| first = M. \| author2 = P. Gronek \| author3 = K. Malarz \| title = Efficient space virtualisation for Hoshen–Kopelman algorithm \| journal = International Journal of Modern Physics C \| volume = 30 \| pages = 1950055–1950099 \| arxiv=1803.09504 \| year = 2019 \| issue = 8 \| doi = 10.1142/S0129183119500554 \| bibcode = 2019IJMPC..3050055K \| s2cid = 4418563 }} 0.1968904(65), 0.19688561(3) \| 0.1600(1), 0.16005(15), 0.1601314(13), 0.160130(3), 0.1601310(10),{{cite journal \| last = Dammer \| first = Stephan M \|author2=Haye Hinrichsen \| title = Spreading with immunization in high dimensions \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2004 \| issue = 7 \| year = 2004 \| pages = P07011 \| doi = 10.1088/1742-5468/2004/07/P07011 \|arxiv = cond-mat/0405577 \|bibcode = 2004JSMTE..07..011D \| s2cid = 118981083 }} 0.1601312(2), 0.16013122(6)
5 \| 10 \| 0.141(1),0.198(1) 0.141(3), 0.1407966(15), 0.1407966(26), 0.14079633(4) \| 0.1181(1),{{Cite journal \|last1=Harris \|first1=A. B. \|last2=Fisch \|first2=R. \|date=1977 \|title=Critical Behavior of Random Resistor Networks \|url=https://link.aps.org/doi/10.1103/PhysRevLett.38.796 \|journal=Physical Review Letters \|volume=38 \|issue=15 \|pages=796–799 \|doi=10.1103/PhysRevLett.38.796\|bibcode=1977PhRvL..38..796H }} 0.118(1), 0.11819(4), 0.118172(1), 0.1181718(3) 0.11817145(3)
6 \| 12 \| 0.106(1), 0.108(3), 0.109017(2), 0.1090117(30), 0.109016661(8) \| 0.0943(1), 0.0942(1), 0.0942019(6), 0.09420165(2)
7 \| 14 \| 0.05950(5), 0.088939(20),{{cite journal \| last = Stauffer \| first = Dietrich \|author2=Robert M. Ziff \| title = Reexamination of Seven-Dimensional Site Percolation Thresholds \| journal = International Journal of Modern Physics C \| volume = 11 \| issue = 1 \| year = 1999 \| pages = 205–209 \| doi = 10.1142/S0129183100000183 \|arxiv = cond-mat/9911090 \|bibcode = 2000IJMPC..11..205S \| s2cid = 119362011 }} 0.0889511(9), 0.0889511(90),{{cite journal \| last = Koza \| first = Zbigniew \|author2= Jakub Poła \| title = From discrete to continuous percolation in dimensions 3 to 7 \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2016 \| issue = 10 \| year = 2016 \| pages = 103206 \| doi = 10.1088/1742-5468/2016/10/103206\| arxiv =1606.08050\| bibcode =2016JSMTE..10.3206K\| s2cid = 118580056 }} 0.088951121(1), \| 0.0787(1), 0.078685(30), 0.0786752(3), 0.078675230(2)
8 \| 16 \| 0.0752101(5), 0.075210128(1) \| 0.06770(5), 0.06770839(7), 0.0677084181(3)
9 \| 18 \| 0.0652095(3), 0.0652095348(6) \| 0.05950(5),{{cite journal \| last = Adler \| first = Joan \| author2 = Yigal Meir \| author3 = Amnon Aharony \| author4 = A. B. Harris \| title = Series Study of Percolation Moments in General Dimension \| journal = Physical Review B \| volume = 41 \| issue = 13 \| year = 1990 \| pages = 9183–9206 \| doi = 10.1103/PhysRevB.41.9183\| pmid = 9993262 \| bibcode = 1990PhRvB..41.9183A \| url = https://repository.upenn.edu/physics_papers/368 }} 0.05949601(5), 0.0594960034(1)
10 \| 20 \| 0.0575930(1), 0.0575929488(4) \| 0.05309258(4), 0.0530925842(2)
11 \| 22 \| 0.05158971(8), 0.0515896843(2) \| 0.04794969(1), 0.04794968373(8)
12 \| 24 \| 0.04673099(6), 0.0467309755(1) \| 0.04372386(1), 0.04372385825(10)
13 \| 26 \| 0.04271508(8), 0.04271507960(10){{Cite journal \| last = Mertens \| first = Stephan \|author2=Christopher Moore \| title = Percolation Thresholds and Fisher Exponents in Hypercubic Lattices \| journal = Physical Review E \| volume = 98 \| issue = 2 \| pages = 022120 \| year = 2018 \|arxiv = 1806.08067\| doi = 10.1103/PhysRevE.98.022120 \| pmid = 30253462 \| bibcode = 2018PhRvE..98b2120M \| s2cid = 52821851 }} \| 0.04018762(1), 0.04018761703(6)

class="wikitable" border="1"

! z

! Site thresholds

! Bond thresholds

| 8

| 0.198(1){{cite journal

| last = Kirkpatrick

| first = Scott

| title = Percolation phenomena in higher dimensions: Approach to the mean-field limit

| journal = Physical Review Letters

| volume = 36

| issue = 2

| year = 1976

| pages = 69–72

| doi = 10.1103/PhysRevLett.36.69| bibcode = 1976PhRvL..36...69K

}}

0.197(6), 0.1968861(14),{{cite journal

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| volume = 67

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| year = 2003

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|arxiv = cond-mat/0202144 |bibcode = 2003PhRvE..67c6101G | s2cid = 43707822

}}

0.196889(3),{{cite journal

| last = Paul

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|author2=Robert M. Ziff |author3=H. Eugene Stanley

| title = Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions

| journal = Physical Review E

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}}

0.196901(5),{{cite journal

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| journal = Phys. Lett. B

| volume = 400

| issue = 3–4

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| doi = 10.1016/S0370-2693(97)00337-7 |arxiv = hep-lat/9612024 |bibcode = 1997PhLB..400..346B | s2cid = 10242417

}}

0.19680(23),

{{cite journal

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| author2 = P. Gronek

| author3 = K. Malarz

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| journal = International Journal of Modern Physics C

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| arxiv=1803.09504

| year = 2019

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| doi = 10.1142/S0129183119500554

| bibcode = 2019IJMPC..3050055K

| s2cid = 4418563

}}

0.1968904(65), 0.19688561(3)

| 0.1600(1), 0.16005(15), 0.1601314(13), 0.160130(3), 0.1601310(10),{{cite journal

| last = Dammer

| first = Stephan M

|author2=Haye Hinrichsen

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| volume = 2004

| issue = 7

| year = 2004

| pages = P07011

| doi = 10.1088/1742-5468/2004/07/P07011 |arxiv = cond-mat/0405577 |bibcode = 2004JSMTE..07..011D | s2cid = 118981083

}}

0.1601312(2), 0.16013122(6)

| 10

| 0.141(1),0.198(1) 0.141(3), 0.1407966(15), 0.1407966(26), 0.14079633(4)

| 0.1181(1),{{Cite journal |last1=Harris |first1=A. B. |last2=Fisch |first2=R. |date=1977 |title=Critical Behavior of Random Resistor Networks |url=https://link.aps.org/doi/10.1103/PhysRevLett.38.796 |journal=Physical Review Letters |volume=38 |issue=15 |pages=796–799 |doi=10.1103/PhysRevLett.38.796|bibcode=1977PhRvL..38..796H }} 0.118(1), 0.11819(4), 0.118172(1), 0.1181718(3) 0.11817145(3)

| 12

| 0.106(1), 0.108(3), 0.109017(2), 0.1090117(30), 0.109016661(8)

| 0.0943(1), 0.0942(1), 0.0942019(6), 0.09420165(2)

| 14

| 0.05950(5), 0.088939(20),{{cite journal

| last = Stauffer

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| title = Reexamination of Seven-Dimensional Site Percolation Thresholds

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| year = 1999

| pages = 205–209

| doi = 10.1142/S0129183100000183 |arxiv = cond-mat/9911090 |bibcode = 2000IJMPC..11..205S | s2cid = 119362011

}}

0.0889511(9), 0.0889511(90),{{cite journal

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| first = Zbigniew

|author2= Jakub Poła

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| journal = Journal of Statistical Mechanics: Theory and Experiment

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| issue = 10

| year = 2016

| pages = 103206

| doi = 10.1088/1742-5468/2016/10/103206| arxiv =1606.08050| bibcode =2016JSMTE..10.3206K| s2cid = 118580056

}}

0.088951121(1),

| 0.0787(1), 0.078685(30), 0.0786752(3), 0.078675230(2)

| 16

| 0.0752101(5), 0.075210128(1)

| 0.06770(5), 0.06770839(7), 0.0677084181(3)

| 18

| 0.0652095(3), 0.0652095348(6)

| 0.05950(5),{{cite journal

| last = Adler

| first = Joan

| author2 = Yigal Meir

| author3 = Amnon Aharony

| author4 = A. B. Harris

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| journal = Physical Review B

| volume = 41

| issue = 13

| year = 1990

| pages = 9183–9206

| doi = 10.1103/PhysRevB.41.9183| pmid = 9993262

| bibcode = 1990PhRvB..41.9183A

| url = https://repository.upenn.edu/physics_papers/368

}} 0.05949601(5), 0.0594960034(1)

| 20

| 0.0575930(1), 0.0575929488(4)

| 0.05309258(4), 0.0530925842(2)

| 22

| 0.05158971(8), 0.0515896843(2)

| 0.04794969(1), 0.04794968373(8)

| 24

| 0.04673099(6), 0.0467309755(1)

| 0.04372386(1), 0.04372385825(10)

| 26

| 0.04271508(8), 0.04271507960(10){{Cite journal

| last = Mertens

| first = Stephan

|author2=Christopher Moore

| title = Percolation Thresholds and Fisher Exponents in Hypercubic Lattices

| journal = Physical Review E

| volume = 98

| issue = 2

| pages = 022120

| year = 2018

|arxiv = 1806.08067| doi = 10.1103/PhysRevE.98.022120

| pmid = 30253462

| bibcode = 2018PhRvE..98b2120M

| s2cid = 52821851

}}

| 0.04018762(1), 0.04018761703(6)

For thresholds on high dimensional hypercubic lattices, we have the asymptotic series expansions

{{cite journal |last1=Gaunt |first1=D. S. |last2=Sykes |first2=M. F. |last3=Ruskin |first3=Heather |title=Percolation processes in d-dimensions |journal=J. Phys. A: Math. Gen. |date=1976 |volume=9 |issue=11 |pages=1899–1911 |doi=10.1088/0305-4470/9/11/015 |bibcode=1976JPhA....9.1899G}}

{{cite journal |last1=Gaunt |first1=D. S. |last2=Ruskin |first2=Heather |title=Bond percolation processes in d-dimensions |journal=J. Phys. A: Math. Gen. |date=1978 |volume=11 |issue=7 |pages=1369 |doi=10.1088/0305-4470/11/7/025|bibcode=1978JPhA...11.1369G}}

{{cite journal |last1=Mertens |first1=Stephan |last2=Moore |first2=Christopher |title=Series Expansion of Critical Densities for Percolation on ℤ^d |journal=J. Phys. A: Math. Theor. |volume=51 |issue=47 |pages=475001 |year=2018 |arxiv=1805.02701 |doi=10.1088/1751-8121/aae65c |s2cid=119399128}}

$p_c^\mathrm{site}(d)=\sigma^{-1}+\frac{3}{2}\sigma^{-2}+\frac{15}{4}\sigma^{-3}+\frac{83}{4}\sigma^{-4}+\frac{6577}{48}\sigma^{-5}+\frac{119077}{96}\sigma^{-6}+{\mathcal O}(\sigma^{-7})$

$p_c^\mathrm{bond}(d)=\sigma^{-1}+\frac{5}{2}\sigma^{-3}+\frac{15}{2}\sigma^{-4}+57\sigma^{-5}+\frac{4855}{12}\sigma^{-6}+{\mathcal O}(\sigma^{-7})$

where $\sigma = 2 d - 1$ . For 13-dimensional bond percolation, for example, the error with the measured value is less than 10⁻⁶, and these formulas can be useful for higher-dimensional systems.

=Thresholds in other higher-dimensional lattices=

class="wikitable" border="1"
d ! lattice ! z ! Site thresholds ! Bond thresholds
4 \|diamond \|5 \|0.2978(2) \|0.2715(3)
4 \|kagome \|8 \|0.2715(3){{cite journal \| last = van der Marck \| first = Steven C. \| title = Site percolation and random walks on d-dimensional Kagome lattices \| journal = Journal of Physics A \| volume = 31 \| issue = 15 \| year = 1998 \| pages = 3449–3460 \| doi = 10.1088/0305-4470/31/15/010 \|arxiv =cond-mat/9801112 \|bibcode = 1998JPhA...31.3449V \| s2cid = 18989583 }} \| 0.177(1)
4 \|bcc \|16 \|0.1037(3) \|0.074(1), 0.074212(1){{cite journal \| last = Xun \| first = Zhipeng \| title = Precise bond percolation thresholds on several four-dimensional lattices \| journal = Physical Review Research \| volume = 2 \| issue = 1 \| year = 2020 \| pages = 013067 \| doi = 10.1103/PhysRevResearch.2.013067\| bibcode = 2020PhRvR...2a3067X \| arxiv = 1910.11408 \| s2cid = 204915841 }}
4 \|fcc, D₄, hypercubic 2NN \|24 \|0.0842(3), 0.08410(23), 0.0842001(11) \|0.049(1), 0.049517(1), 0.0495193(8)
4 \|hypercubic NN+2NN \|32 \|0.06190(23), 0.0617731(19){{cite journal \| last = Zhao \| first = Pengyu \| author2 = Jinhong Yan \| author3 = Zhipeng Xun \| author4 = Dapeng Hao \| author5 = Robert M. Ziff \| title = Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2022 \| issue = 3 \| year = 2022 \| pages = 033202 \| doi = 10.1088/1742-5468/ac52a8 \| arxiv = 2109.11195 \| bibcode = 2022JSMTE2022c3202Z \| s2cid = 237605083 }} \|0.035827(1), 0.0338047(27)
4 \|hypercubic 3NN \|32 \|0.04540(23) \|
4 \|hypercubic NN+3NN \|40 \|0.04000(23) \| 0.0271892(22)
4 \|hypercubic 2NN+3NN \|56 \|0.03310(23) \|0.0194075(15)
4 \|hypercubic NN+2NN+3NN \|64 \|0.03190(23), 0.0319407(13) \|0.0171036(11)
4 \|hypercubic NN+2NN+3NN+4NN \|88 \| 0.0231538(12) \| 0.0122088(8)
4 \|hypercubic NN+...+5NN \| 136 \| 0.0147918(12) \| 0.0077389(9)
4 \|hypercubic NN+...+6NN \| 232 \| 0.0088400(10) \| 0.0044656(11)
4 \|hypercubic NN+...+7NN \| 296 \| 0.0070006(6) \| 0.0034812(7)
4 \|hypercubic NN+...+8NN \| 320 \| 0.0064681(9) \| 0.0032143(8)
4 \|hypercubic NN+...+9NN \| 424 \| 0.0048301(9) \| 0.0024117(7)
5 \|diamond \|6 \|0.2252(3) \|0.2084(4)
5 \|kagome \|10 \|0.2084(4) \|0.130(2)
5 \|bcc \|32 \|0.0446(4) \|0.033(1)
5 \|fcc, D₅, hypercubic 2NN \|40 \|0.0431(3), 0.0435913(6) \| 0.026(2), 0.0271813(2)
5 \|hypercubic NN+2NN \|50 \|0.0334(2){{cite journal \| last = Löbl \| first = Matthias C. \| title = Loss-tolerant architecture for quantum computing with quantum emitters \| journal = Quantum \| volume = 8 \| pages = 1302 \| year = 2024 \| doi = 10.22331/q-2024-03-28-1302 \| arxiv = 2304.03796 }} \|0.0213(1)
6 \|diamond \|7 \|0.1799(5) \|0.1677(7)
6 \|kagome \|12 \|0.1677(7) \|
6 \|fcc, D₆ \|60 \|0.0252(5), 0.02602674(12) \|0.01741556(5)
6 \|bcc \|64 \|0.0199(5) \|
6 \|E₆ \|72 \|0.02194021(14) \|0.01443205(8)
7 \|fcc, D₇ \|84 \|0.01716730(5) \|0.012217868(13)
7 \|E₇ \| 126 \|0.01162306(4) \|0.00808368(2)
8 \|fcc, D₈ \|112 \|0.01215392(4) \|0.009081804(6)
8 \|E₈ \| 240 \|0.00576991(2) \|0.004202070(2)
9 \|fcc, D₉ \|144 \|0.00905870(2) \|0.007028457(3)
9 \| $\Lambda_9$ \| 272 \|0.00480839(2) \|0.0037006865(11)
10 \|fcc, D₁₀ \|180 \|0.007016353(9) \|0.005605579(6)
11 \|fcc, D₁₁ \|220 \|0.005597592(4) \|0.004577155(3)
12 \|fcc, D₁₂ \|264 \|0.004571339(4) \|0.003808960(2)
13 \|fcc, D₁₃ \|312 \|0.003804565(3) \|0.0032197013(14)

class="wikitable" border="1"

! lattice

! z

! Site thresholds

! Bond thresholds

|diamond

|0.2978(2)

|0.2715(3)

|kagome

|0.2715(3){{cite journal

| last = van der Marck

| first = Steven C.

| title = Site percolation and random walks on d-dimensional Kagome lattices

| journal = Journal of Physics A

| volume = 31

| issue = 15

| year = 1998

| pages = 3449–3460

| doi = 10.1088/0305-4470/31/15/010 |arxiv =cond-mat/9801112 |bibcode = 1998JPhA...31.3449V | s2cid = 18989583

}}

| 0.177(1)

|bcc

|16

|0.1037(3)

|0.074(1), 0.074212(1){{cite journal

| last = Xun

| first = Zhipeng

| title = Precise bond percolation thresholds on several four-dimensional lattices

| journal = Physical Review Research

| volume = 2

| issue = 1

| year = 2020

| pages = 013067

| doi = 10.1103/PhysRevResearch.2.013067| bibcode = 2020PhRvR...2a3067X

| arxiv = 1910.11408

| s2cid = 204915841

}}

|fcc, D₄, hypercubic 2NN

|24

|0.0842(3), 0.08410(23), 0.0842001(11)

|0.049(1), 0.049517(1), 0.0495193(8)

|hypercubic NN+2NN

|32

|0.06190(23), 0.0617731(19){{cite journal

| last = Zhao

| first = Pengyu

| author2 = Jinhong Yan

| author3 = Zhipeng Xun

| author4 = Dapeng Hao

| author5 = Robert M. Ziff

| title = Site and bond percolation on four-dimensional simple hypercubic lattices with extended neighborhoods

| journal = Journal of Statistical Mechanics: Theory and Experiment

| volume = 2022

| issue = 3

| year = 2022

| pages = 033202

| doi = 10.1088/1742-5468/ac52a8

| arxiv = 2109.11195

| bibcode = 2022JSMTE2022c3202Z

| s2cid = 237605083

}}

|0.035827(1), 0.0338047(27)

|hypercubic 3NN

|32

|0.04540(23)

|hypercubic NN+3NN

|40

|0.04000(23)

| 0.0271892(22)

|hypercubic 2NN+3NN

|56

|0.03310(23)

|0.0194075(15)

|hypercubic NN+2NN+3NN

|64

|0.03190(23), 0.0319407(13)

|0.0171036(11)

|hypercubic NN+2NN+3NN+4NN

|88

| 0.0231538(12)

| 0.0122088(8)

|hypercubic NN+...+5NN

| 136

| 0.0147918(12)

| 0.0077389(9)

|hypercubic NN+...+6NN

| 232

| 0.0088400(10)

| 0.0044656(11)

|hypercubic NN+...+7NN

| 296

| 0.0070006(6)

| 0.0034812(7)

|hypercubic NN+...+8NN

| 320

| 0.0064681(9)

| 0.0032143(8)

|hypercubic NN+...+9NN

| 424

| 0.0048301(9)

| 0.0024117(7)

|diamond

|0.2252(3)

|0.2084(4)

|kagome

|10

|0.2084(4)

|0.130(2)

|bcc

|32

|0.0446(4)

|0.033(1)

|fcc, D₅, hypercubic 2NN

|40

|0.0431(3), 0.0435913(6)

| 0.026(2), 0.0271813(2)

|hypercubic NN+2NN

|50

|0.0334(2){{cite journal

| last = Löbl

| first = Matthias C.

| title = Loss-tolerant architecture for quantum computing with quantum emitters

| journal = Quantum

| volume = 8

| pages = 1302

| year = 2024

| doi = 10.22331/q-2024-03-28-1302

| arxiv = 2304.03796

}}

|0.0213(1)

|diamond

|0.1799(5)

|0.1677(7)

|kagome

|12

|0.1677(7)

|fcc, D₆

|60

|0.0252(5), 0.02602674(12)

|0.01741556(5)

|bcc

|64

|0.0199(5)

|E₆

|72

|0.02194021(14)

|0.01443205(8)

|fcc, D₇

|84

|0.01716730(5)

|0.012217868(13)

|E₇

| 126

|0.01162306(4)

|0.00808368(2)

|fcc, D₈

|112

|0.01215392(4)

|0.009081804(6)

|E₈

| 240

|0.00576991(2)

|0.004202070(2)

|fcc, D₉

|144

|0.00905870(2)

|0.007028457(3)

| $\Lambda_9$

| 272

|0.00480839(2)

|0.0037006865(11)

|fcc, D₁₀

|180

|0.007016353(9)

|0.005605579(6)

|fcc, D₁₁

|220

|0.005597592(4)

|0.004577155(3)

|fcc, D₁₂

|264

|0.004571339(4)

|0.003808960(2)

|fcc, D₁₃

|312

|0.003804565(3)

|0.0032197013(14)

= Thresholds in one-dimensional long-range percolation =

File:LR 1d clusters wiki.png

In a one-dimensional chain we establish bonds between distinct sites $i$ and $j$ with probability $p=\frac{C}$

i-j|^{1+\sigma}} decaying as a power-law with an exponent

\sigma>0

. Percolation occurs{{Cite journal|last=Schulman|first=L. S.|date=1983|title=Long range percolation in one dimension|journal=Journal of Physics A: Mathematical and General|language=en|volume=16|issue=17|pages=L639–L641|doi=10.1088/0305-4470/16/17/001|issn=0305-4470|bibcode=1983JPhA...16L.639S}}{{Cite journal|last1=Aizenman|first1=M.|last2=Newman|first2=C. M.|date=1986-12-01|title=Discontinuity of the percolation density in one dimensional 1/{{!}}x−y{{!}}2 percolation models|journal=Communications in Mathematical Physics|language=en|volume=107|issue=4|pages=611–647|doi=10.1007/BF01205489|issn=0010-3616|bibcode=1986CMaPh.107..611A|s2cid=117904292}} at a critical value

C_c<1

for

\sigma<1

. The numerically determined percolation thresholds are given by:

{| class="wikitable"

! $\sigma$

! $C_c$

|Critical thresholds $C_c$ as a function of $\sigma$ .{{Cite journal |last1=Gori|first1=G.|last2=Michelangeli|first2=M.|last3=Defenu|first3=N.|last4=Trombettoni|first4=A. |date=2017|title=One-dimensional long-range percolation: A numerical study|journal=Physical Review E |volume=96|issue=1|pages=012108|pmid=29347133|doi=10.1103/physreve.96.012108|arxiv=1610.00200|s2cid=9926800 |bibcode=2017PhRvE..96a2108G}}
The dotted line is the rigorous lower bound.

0.1

|0.047685(8)

|rowspan=9|300x300px

0.2

|0.093211(16)

0.3

|0.140546(17)

0.4

|0.193471(15)

0.5

|0.25482(5)

0.6

|0.327098(6)

0.7

|0.413752(14)

0.8

|0.521001(14)

0.9

|0.66408(7)

= Thresholds on hyperbolic, hierarchical, and tree lattices =

In these lattices there may be two percolation thresholds: the lower threshold is the probability above which infinite clusters appear, and the upper is the probability above which there is a unique infinite cluster.

[[File:TriangularHyperbolic.jpg|thumb|Visualization of a triangular hyperbolic lattice {3,7} projected on the Poincaré disk (red bonds). Green bonds show dual-clusters on the {7,3} lattice{{cite journal

| last = Baek

| first = S.K.

| author2 = Petter Minnhagen |author3=Beom Jun Kim

| title = Comment on 'Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees'

| journal = J. Phys. A: Math. Theor.

| volume = 42

| issue = 47

| pages = 478001

| doi = 10.1088/1751-8113/42/47/478001

| year = 2009|bibcode = 2009JPhA...42U8001B |arxiv = 0910.4340 | s2cid = 102489139

}}

]]

File:Hn-np.jpg

class="wikitable"
Lattice ! z \| $\overline z$ ! colspan="2"\| Site percolation threshold ! colspan="2"\| Bond percolation threshold
\| \| \|Lower \|Upper \|Lower \|Upper
{3,7} hyperbolic \| 7 \|7 \| 0.26931171(7), 0.20 \| 0.73068829(7), 0.73(2){{cite journal \| last = Lopez \| first = Jorge H. \| author2 = J. M. Schwarz \| author2-link = Jennifer Schwarz \| title = Constraint percolation on hyperbolic lattices \| journal = Physical Review E \| volume = 96 \| issue = 5 \| pages = 052108 \| doi = 10.1103/PhysRevE.96.052108 \| pmid = 29347694 \| year = 2017 \|arxiv = 1512.05404\| bibcode = 2017PhRvE..96e2108L \| s2cid = 44770310 }} \| 0.20, 0.1993505(5) \| 0.37, 0.4694754(8)
{3,8} hyperbolic \| 8 \| 8 \| 0.20878618(9) \| 0.79121382(9) \| 0.1601555(2) \| 0.4863559(6)
{3,9} hyperbolic \| 9 \| 9 \| 0.1715770(1) \| 0.8284230(1) \| 0.1355661(4) \| 0.4932908(1)
{4,5} hyperbolic \| 5 \|5 \| 0.29890539(6) \| 0.8266384(5) \|0.27,{{cite journal \| last = Baek \| first = S.K. \| author2 = Petter Minnhagen \|author3=Beom Jun Kim \| title = Percolation on hyperbolic lattices \| journal = Physical Review E \| volume = 79 \| issue = 1 \| pages = 011124 \| doi = 10.1103/PhysRevE.79.011124 \| year = 2009 \| pmid = 19257018\|bibcode = 2009PhRvE..79a1124B \|arxiv = 0901.0483 \| s2cid = 29468086 }} 0.2689195(3){{cite journal \| last = Mertens \| first = Stephan \| author2 = Cristopher Moore \| title = Percolation thresholds in hyperbolic lattices \| journal = Physical Review E \| volume = 96 \| issue = 4 \| pages = 042116 \| doi = 10.1103/PhysRevE.96.042116 \| pmid = 29347529 \| year = 2017 \| arxiv = 1708.05876\| bibcode = 2017PhRvE..96d2116M \| s2cid = 39025690 }} \|0.52, 0.6487772(3)
{4,6} hyperbolic \| 6 \| 6 \| 0.22330172(3) \| 0.87290362(7) \| 0.20714787(9) \| 0.6610951(2)
{4,7} hyperbolic \| 7 \| 7 \| 0.17979594(1) \| 0.89897645(3) \| 0.17004767(3) \| 0.66473420(4)
{4,8} hyperbolic \| 8 \| 8 \| 0.151035321(9) \| 0.91607962(7) \| 0.14467876(3) \| 0.66597370(3)
{4,9} hyperbolic \| 8 \| 8 \| 0.13045681(3) \| 0.92820305(3) \| 0.1260724(1) \| 0.66641596(2)
{5,5} hyperbolic \| 5 \|5 \| 0.26186660(5) \| 0.89883342(7) \| 0.263(10), 0.25416087(3) \| 0.749(10) 0.74583913(3)
{7,3} hyperbolic \| 3 \|3 \| 0.54710885(10) \| 0.8550371(5), 0.86(2) \|0.53, 0.551(10), 0.5305246(8) \|0.72, 0.810(10), 0.8006495(5)
{∞,3} Cayley tree \| 3 \|3 \| {{frac\|1\|2}} \| \|{{frac\|1\|2}} \|1
Enhanced binary tree (EBT) \| \| \| \| \|0.304(1), 0.306(10),{{cite journal \| last = Gu \| first = Hang \| author2 = Robert M. Ziff \| title = Crossing on hyperbolic lattices \| journal = Physical Review E \| volume = 85 \| issue = 5 \| pages = 051141 \| doi = 10.1103/PhysRevE.85.051141 \| pmid = 23004737 \| year = 2012\| bibcode = 2012PhRvE..85e1141G\| arxiv = 1111.5626\| s2cid = 7141649 }} ({{radic\|13}} − 3)/2 = 0.302776{{cite journal \| last = Minnhagen \| first = Petter \| author2 = Seung Ki Baek \| title = Analytic results for the percolation transitions of the enhanced binary tree \| journal = Physical Review E \| volume = 82 \| issue = 1 \| pages = 011113 \| doi = 10.1103/PhysRevE.82.011113 \| pmid = 20866571 \| year = 2010\| arxiv = 1003.6012\| bibcode = 2010PhRvE..82a1113M\| s2cid = 21018113 }} \|0.48, 0.564(1),{{cite journal \| last = Nogawa \| first = Tomoaki \|author2=Takehisa Hasegawa \| title = Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees \| journal = J. Phys. A: Math. Theor. \| volume = 42 \| issue = 14 \| pages = 145001 \| doi = 10.1088/1751-8113/42/14/145001 \| year = 2009\|bibcode = 2009JPhA...42n5001N \|arxiv = 0810.1602 \| s2cid = 118367190 }} 0.564(10), {{frac\|1\|2}}
Enhanced binary tree dual \| \| \| \| \|0.436(1), 0.452(10) \|0.696(1), 0.699(10)
Non-Planar Hanoi Network (HN-NP) \| \| \| \| \|0.319445{{cite journal \| last = Boettcher \| first = Stefan \| author2 = Jessica L. Cook \|author3=Robert M. Ziff \| title = Patchy percolation on a hierarchical network with small-world bonds \| journal = Physical Review E \| volume = 80 \| issue = 4 \| pages = 041115 \| doi = 10.1103/PhysRevE.80.041115 \| pmid = 19905281 \| year = 2009\|bibcode = 2009PhRvE..80d1115B \| arxiv = 0907.2717 \| s2cid = 119265110 }} \|0.381996
Cayley tree with grandparents \| \| 8 \| \| \|0.158656326{{cite journal \| last = Kozáková \| first = Iva \| title = Critical percolation of virtually free groups and other tree-like graphs \| journal = Annals of Probability \| volume = 37 \| issue = 6 \| pages = 2262–2296 \| doi = 10.1214/09-AOP458 \| year = 2009\| arxiv = 0801.4153 }} \|

class="wikitable"

Lattice

! z

| $\overline z$

! colspan="2"| Site percolation threshold

! colspan="2"| Bond percolation threshold

|Lower

|Upper

|Lower

|Upper

{3,7} hyperbolic

| 7

| 0.26931171(7), 0.20

| 0.73068829(7), 0.73(2){{cite journal

| last = Lopez

| first = Jorge H.

| author2 = J. M. Schwarz | author2-link = Jennifer Schwarz

| title = Constraint percolation on hyperbolic lattices

| journal = Physical Review E

| volume = 96

| issue = 5

| pages = 052108

| doi = 10.1103/PhysRevE.96.052108

| pmid = 29347694

| year = 2017

|arxiv = 1512.05404| bibcode = 2017PhRvE..96e2108L

| s2cid = 44770310

}}

| 0.20, 0.1993505(5)

| 0.37, 0.4694754(8)

{3,8} hyperbolic

| 8

| 0.20878618(9)

| 0.79121382(9)

| 0.1601555(2)

| 0.4863559(6)

{3,9} hyperbolic

| 9

| 0.1715770(1)

| 0.8284230(1)

| 0.1355661(4)

| 0.4932908(1)

{4,5} hyperbolic

| 5

| 0.29890539(6)

| 0.8266384(5)

|0.27,{{cite journal

| last = Baek

| first = S.K.

| author2 = Petter Minnhagen |author3=Beom Jun Kim

| title = Percolation on hyperbolic lattices

| journal = Physical Review E

| volume = 79

| issue = 1

| pages = 011124

| doi = 10.1103/PhysRevE.79.011124

| year = 2009

| pmid = 19257018|bibcode = 2009PhRvE..79a1124B |arxiv = 0901.0483 | s2cid = 29468086

}}

0.2689195(3){{cite journal

| last = Mertens

| first = Stephan

| author2 = Cristopher Moore

| title = Percolation thresholds in hyperbolic lattices

| journal = Physical Review E

| volume = 96

| issue = 4

| pages = 042116

| doi = 10.1103/PhysRevE.96.042116

| pmid = 29347529

| year = 2017

| arxiv = 1708.05876| bibcode = 2017PhRvE..96d2116M

| s2cid = 39025690

}}

|0.52, 0.6487772(3)

{4,6} hyperbolic

| 6

| 0.22330172(3)

| 0.87290362(7)

| 0.20714787(9)

| 0.6610951(2)

{4,7} hyperbolic

| 7

| 0.17979594(1)

| 0.89897645(3)

| 0.17004767(3)

| 0.66473420(4)

{4,8} hyperbolic

| 8

| 0.151035321(9)

| 0.91607962(7)

| 0.14467876(3)

| 0.66597370(3)

{4,9} hyperbolic

| 8

| 0.13045681(3)

| 0.92820305(3)

| 0.1260724(1)

| 0.66641596(2)

{5,5} hyperbolic

| 5

| 0.26186660(5)

| 0.89883342(7)

| 0.263(10), 0.25416087(3)

| 0.749(10) 0.74583913(3)

{7,3} hyperbolic

| 3

| 0.54710885(10)

| 0.8550371(5), 0.86(2)

|0.53, 0.551(10), 0.5305246(8)

|0.72, 0.810(10), 0.8006495(5)

{∞,3} Cayley tree

| 3

| {{frac|1|2}}

|{{frac|1|2}}

Enhanced binary tree (EBT)

|0.304(1), 0.306(10),{{cite journal

| last = Gu

| first = Hang

| author2 = Robert M. Ziff

| title = Crossing on hyperbolic lattices

| journal = Physical Review E

| volume = 85

| issue = 5

| pages = 051141

| doi = 10.1103/PhysRevE.85.051141

| pmid = 23004737

| year = 2012| bibcode = 2012PhRvE..85e1141G| arxiv = 1111.5626| s2cid = 7141649

}}

({{radic|13}} − 3)/2 = 0.302776{{cite journal

| last = Minnhagen

| first = Petter

| author2 = Seung Ki Baek

| title = Analytic results for the percolation transitions of the enhanced binary tree

| journal = Physical Review E

| volume = 82

| issue = 1

| pages = 011113

| doi = 10.1103/PhysRevE.82.011113

| pmid = 20866571

| year = 2010| arxiv = 1003.6012| bibcode = 2010PhRvE..82a1113M| s2cid = 21018113

}}

|0.48, 0.564(1),{{cite journal

| last = Nogawa

| first = Tomoaki

|author2=Takehisa Hasegawa

| title = Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees

| journal = J. Phys. A: Math. Theor.

| volume = 42

| issue = 14

| pages = 145001

| doi = 10.1088/1751-8113/42/14/145001

| year = 2009|bibcode = 2009JPhA...42n5001N |arxiv = 0810.1602 | s2cid = 118367190

}}

0.564(10), {{frac|1|2}}

Enhanced binary tree dual

|0.436(1), 0.452(10)

|0.696(1), 0.699(10)

Non-Planar Hanoi Network (HN-NP)

|0.319445{{cite journal

| last = Boettcher

| first = Stefan

| author2 = Jessica L. Cook |author3=Robert M. Ziff

| title = Patchy percolation on a hierarchical network with small-world bonds

| journal = Physical Review E

| volume = 80

| issue = 4

| pages = 041115

| doi = 10.1103/PhysRevE.80.041115

| pmid = 19905281

| year = 2009|bibcode = 2009PhRvE..80d1115B | arxiv = 0907.2717

| s2cid = 119265110

}}

|0.381996

Cayley tree with grandparents

| 8

|0.158656326{{cite journal

| last = Kozáková

| first = Iva

| title = Critical percolation of virtually free groups and other tree-like graphs

| journal = Annals of Probability

| volume = 37

| issue = 6

| pages = 2262–2296

| doi = 10.1214/09-AOP458

| year = 2009| arxiv = 0801.4153

}}

Note: {m,n} is the Schläfli symbol, signifying a hyperbolic lattice in which n regular m-gons meet at every vertex

For bond percolation on {P,Q}, we have by duality $p_{c,\ell}(P,Q) + p_{c,u}(Q,P) = 1$ . For site percolation, $p_{c,\ell}(3,Q) + p_{c,u}(3,Q) = 1$ because of the self-matching of triangulated lattices.

Cayley tree (Bethe lattice) with coordination number $z : p_c = 1 / ( z - 1 )$

= Thresholds for directed percolation =

File:(1+1)D Kagome Lattice.png

File:(1+1)D Square Lattice.png

File:(1+1)D Triangular Lattice.png

File:(2+1)D SC Lattice.png

File:(2+1)D BCC Lattice.png

class="wikitable"
Lattice ! z ! width=40% \| Site percolation threshold ! width=40% \| Bond percolation threshold
(1+1)-d honeycomb \| 1.5 \| 0.8399316(2), 0.839933(5),{{cite journal \| last = Jensen \| first = Iwan \| author-link = § \|author2=Anthony J. Guttmann \| title = Series expansions of the percolation probability for directed square and honeycomb lattices \| journal = J. Phys. A: Math. Gen. \| volume = 28 \| issue = 17 \| year = 1995 \| pages = 4813–4833 \| doi = 10.1088/0305-4470/28/17/015 \|arxiv = cond-mat/9509121 \|bibcode = 1995JPhA...28.4813J \| s2cid = 118993303 }} $= \sqrt{p_c({\mathrm{site}})}$ of (1+1)-d sq. \| 0.8228569(2), 0.82285680(6)
(1+1)-d kagome \| 2 \| 0.7369317(2), 0.73693182(4) \| 0.6589689(2), 0.65896910(8)
(1+1)-d square, diagonal \| 2 \| 0.705489(4),{{cite journal \| last = Essam \| first = J. W. \| author2= A. J. Guttmann \| author3 = K. De'Bell \| title = On two-dimensional directed percolation \| journal = J. Phys. A \| volume = 21 \| issue =19 \| year = 1988 \| pages = 3815–3832 \| doi = 10.1088/0305-4470/21/19/018 \| bibcode =1988JPhA...21.3815E}} 0.705489(4),{{cite journal \| last = Lübeck \| first = S. \|author2=R. D. Willmann \| title = Universal scaling behaviour of directed percolation and the pair contact process in an external field \| journal = J. Phys. A \| volume = 35 \| issue =48 \| year = 2002 \| pages = 10205 \| doi = 10.1088/0305-4470/35/48/301\|arxiv = cond-mat/0210403 \|bibcode = 2002JPhA...3510205L \| s2cid = 11831269 }} 0.70548522(4),{{cite journal \| last = Jensen \| first = Iwan \| title = Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice \| journal = J. Phys. A \| volume = 32 \| issue = 28 \| year = 1999 \| pages = 5233–5249 \| doi = 10.1088/0305-4470/32/28/304 \|arxiv = cond-mat/9906036 \|bibcode = 1999JPhA...32.5233J \| s2cid = 2681356 }} 0.70548515(20),{{cite journal \| last = Jensen \| first = Iwan \| title = Low-density series expansions for directed percolation: III. Some two-dimensional lattices \| journal = J. Phys. A: Math. Gen. \| volume = 37 \| issue = 4 \| year = 2004 \| pages = 6899–6915 \| doi = 10.1088/0305-4470/37/27/003 \|arxiv = cond-mat/0405504\|bibcode = 2004JPhA...37.6899J \| citeseerx = 10.1.1.700.2691 \| s2cid = 119326380 }} 0.7054852(3), {{cite journal \| last = Wang \| first = Junfeng \| author2=Zongzheng Zhou \| author3=Qingquan Liu \| author4=Timothy M. Garoni \| author5=Youjin Deng \| title = A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions \| arxiv = 1201.3006 \| doi=10.1103/PhysRevE.88.042102 \| pmid = 24229111 \| volume=88 \| issue = 4 \| pages = 042102 \| journal=Physical Review E \| bibcode=2013PhRvE..88d2102W \| year = 2013 \| s2cid = 43011467 }} \| 0.644701(2),{{cite journal \| last = Essam \| first = John \|author2=K. De'Bell \|author3=J. Adler \|author3-link= Joan Adler \|author4=F. M. Bhatti \| title = Analysis of extended series for bond percolation on the directed square lattice \| journal = Physical Review B \| volume = 33 \| issue = 2 \| year = 1986 \| pages = 1982–1986 \| doi = 10.1103/PhysRevB.33.1982\| pmid = 9938508 \|bibcode = 1986PhRvB..33.1982E }} 0.644701(1),{{cite journal \| last = Baxter \| first = R. J. \|author2=A. J. Guttmann \| title = Series expansion of the percolation probability for the directed square lattice \| journal = J. Phys. A \| volume = 21 \| issue = 15 \| year = 1988 \| pages = 3193–3204 \| doi = 10.1088/0305-4470/21/15/008\|bibcode = 1988JPhA...21.3193B }} 0.644701(1), 0.6447006(10), 0.64470015(5),{{cite journal \| last = Jensen \| first = Iwan \| title = Low-density series expansions for directed percolation on square and triangular lattices \| journal = J. Phys. A \| volume = 29 \| issue = 22 \| year = 1996 \| pages = 7013–7040 \| doi = 10.1088/0305-4470/29/22/007\|bibcode = 1996JPhA...29.7013J \| s2cid = 121332666 }} 0.644700185(5), 0.6447001(2), 0.643(2)
(1+1)-d triangular \| 3 \| 0.595646(3), 0.5956468(5), 0.5956470(3) \| 0.478018(2), 0.478025(1), 0.4780250(4) 0.479(3)
(2+1)-d simple cubic, diagonal planes \| 3 \| 0.43531(1), 0.43531411(10) \| 0.382223(7), 0.38222462(6) 0.383(3)
(2+1)-d square nn (= bcc) \| 4 \| 0.3445736(3),{{cite journal \| last = Grassberger \| first = P. \| title = Local persistence in directed percolation \| journal = Journal of Statistical Mechanics: Theory and Experiment \| volume = 2009 \| issue = 8 \| year = 2009 \| pages = P08021 \| doi = 10.1088/1742-5468/2009/08/P08021 \|bibcode = 2009JSMTE..08..021G \|arxiv = 0907.4021 \| s2cid = 119236556 }} 0.344575(15) 0.3445740(2) \| 0.2873383(1),{{cite journal \| last = Perlsman \| first = E. \|author2=S. Havlin \| title = Method to estimate critical exponents using numerical studies \| journal = Europhys. Lett. \| volume = 58 \| issue = 2 \| year = 2002 \| pages = 176–181 \| url =https://semanticscholar.org/paper/502191a5395e90141d8f22da5576825a693cda5a \| doi = 10.1209/epl/i2002-00621-7\|bibcode = 2002EL.....58..176P \| s2cid = 67818664 }} 0.287338(3){{cite journal \| last = Grassberger \| first = P. \|author2=Y.-C. Zhang \| title = "Self-organized" formulation of standard percolation phenomena \| journal = Physica A \| volume = 224 \| issue = 1 \| year = 1996 \| pages = 169–179 \| doi = 10.1016/0378-4371(95)00321-5\|bibcode = 1996PhyA..224..169G }} 0.28733838(4) 0.287(3)
(2+1)-d fcc \| \| \| 0.199(2))
(3+1)-d hypercubic, diagonal \| 4 \| 0.3025(10),{{cite journal \| last = Adler \| first = Joan \| author-link = Joan Adler \| author2 = J. Berger \|author3=M. A. M. S. Duarte \|author4=Y. Meir \| title = Directed percolation in 3+1 dimensions \| journal = Physical Review B \| volume = 37 \| issue = 13 \| year = 1988 \| pages = 7529–7533 \| doi = 10.1103/PhysRevB.37.7529\| pmid = 9944046 \|bibcode = 1988PhRvB..37.7529A }} 0.30339538(5) \| 0.26835628(5), 0.2682(2)
(3+1)-d cubic, nn \| 6 \|0.2081040(4) \| 0.1774970(5)
(3+1)-d bcc \| 8 \|0.160950(30), 0.16096128(3) \|0.13237417(2)
(4+1)-d hypercubic, diagonal \| 5 \| 0.23104686(3) \| 0.20791816(2), 0.2085(2){{cite journal \| last = Blease \| first = J. \| title = Series expansions for the directed-bond percolation problem \| journal = J. Phys. C: Solid State Phys. \| volume = 10 \| issue = 7 \| year = 1977 \| pages = 917–924 \| doi = 10.1088/0022-3719/10/7/003\| bibcode = 1977JPhC...10..917B}}
(4+1)-d hypercubic, nn \| 8 \| 0.1461593(2), 0.1461582(3){{cite journal \| last = Grassberger \| first = Peter \| title = Logarithmic corrections in (4 + 1)-dimensional directed percolation \| journal = Physical Review E \| volume = 79 \| issue = 5 \| year = 2009 \| pages = 052104 \| doi = 10.1103/PhysRevE.79.052104\| pmid = 19518501 \|bibcode = 2009PhRvE..79e2104G \|arxiv = 0904.0804 \| s2cid = 23876626 }} \| 0.1288557(5)
(4+1)-d bcc \| 16 \| 0.075582(17),{{cite journal \| last = Lübeck \| first = S. \|author2=R. D. Willmann \| title = Universal scaling behavior of directed percolation around the upper critical dimension \| journal = J. Stat. Phys. \| volume = 115 \| issue = 5–6 \| year = 2004 \| pages = 1231–1250 \| doi = 10.1023/B:JOSS.0000028059.24904.3b\|arxiv = cond-mat/0401395 \|bibcode = 2004JSP...115.1231L \| citeseerx = 10.1.1.310.8700 \| s2cid = 16267627 }} 0.0755850(3), 0.07558515(1) \| 0.063763395(5)
(5+1)-d hypercubic, diagonal \| 6 \| 0.18651358(2) \| 0.170615155(5), 0.1714(1)
(5+1)-d hypercubic, nn \| 10 \| 0.1123373(2) \| 0.1016796(5)
(5+1)-d hypercubic bcc \| 32 \|0.035967(23), 0.035972540(3) \| 0.0314566318(5)
(6+1)-d hypercubic, diagonal \| 7 \| 0.15654718(1) \| 0.145089946(3), 0.1458
(6+1)-d hypercubic, nn \| 12 \| 0.0913087(2) \| 0.0841997(14)
(6+1)-d hypercubic bcc \| 64 \| 0.017333051(2) \| 0.01565938296(10)
(7+1)-d hypercubic, diagonal \| 8 \| 0.135004176(10) \| 0.126387509(3), 0.1270(1)
(7+1)-d hypercubic,nn \| 14 \| 0.07699336(7) \| 0.07195(5)
(7+1)-d bcc \| 128 \| 0.008 432 989(2) \| 0.007 818 371 82(6)

class="wikitable"

Lattice

! z

! width=40% | Site percolation threshold

! width=40% | Bond percolation threshold

(1+1)-d honeycomb

| 1.5

| 0.8399316(2), 0.839933(5),{{cite journal

| last = Jensen

| first = Iwan

| author-link = §

|author2=Anthony J. Guttmann

| title = Series expansions of the percolation probability for directed square and honeycomb lattices

| journal = J. Phys. A: Math. Gen.

| volume = 28

| issue = 17

| year = 1995

| pages = 4813–4833

| doi = 10.1088/0305-4470/28/17/015 |arxiv = cond-mat/9509121 |bibcode = 1995JPhA...28.4813J | s2cid = 118993303

}} $= \sqrt{p_c({\mathrm{site}})}$ of (1+1)-d sq.

| 0.8228569(2), 0.82285680(6)

(1+1)-d kagome

| 2

| 0.7369317(2), 0.73693182(4)

| 0.6589689(2), 0.65896910(8)

(1+1)-d square, diagonal

| 2

| 0.705489(4),{{cite journal

| last = Essam

| first = J. W.

| author2= A. J. Guttmann

| author3 = K. De'Bell

| title = On two-dimensional directed percolation

| journal = J. Phys. A

| volume = 21

| issue =19

| year = 1988

| pages = 3815–3832

| doi = 10.1088/0305-4470/21/19/018 | bibcode =1988JPhA...21.3815E}}

0.705489(4),{{cite journal

| last = Lübeck

| first = S.

|author2=R. D. Willmann

| title = Universal scaling behaviour of directed percolation and the pair contact process in an external field

| journal = J. Phys. A

| volume = 35

| issue =48

| year = 2002

| pages = 10205

| doi = 10.1088/0305-4470/35/48/301|arxiv = cond-mat/0210403 |bibcode = 2002JPhA...3510205L | s2cid = 11831269

}}

0.70548522(4),{{cite journal

| last = Jensen

| first = Iwan

| title = Low-density series expansions for directed percolation: I. A new efficient algorithm with applications to the square lattice

| journal = J. Phys. A

| volume = 32

| issue = 28

| year = 1999

| pages = 5233–5249

| doi = 10.1088/0305-4470/32/28/304 |arxiv = cond-mat/9906036 |bibcode = 1999JPhA...32.5233J | s2cid = 2681356

}} 0.70548515(20),{{cite journal

| last = Jensen

| first = Iwan

| title = Low-density series expansions for directed percolation: III. Some two-dimensional lattices

| journal = J. Phys. A: Math. Gen.

| volume = 37

| issue = 4

| year = 2004

| pages = 6899–6915

| doi = 10.1088/0305-4470/37/27/003 |arxiv = cond-mat/0405504|bibcode = 2004JPhA...37.6899J | citeseerx = 10.1.1.700.2691

| s2cid = 119326380

}} 0.7054852(3),

{{cite journal

| last = Wang | first = Junfeng

| author2=Zongzheng Zhou

| author3=Qingquan Liu

| author4=Timothy M. Garoni

| author5=Youjin Deng

| title = A high-precision Monte Carlo study of directed percolation in (d + 1) dimensions

| arxiv = 1201.3006

| doi=10.1103/PhysRevE.88.042102

| pmid = 24229111

| volume=88

| issue = 4

| pages = 042102

| journal=Physical Review E

| bibcode=2013PhRvE..88d2102W

| year = 2013

| s2cid = 43011467

}}

| 0.644701(2),{{cite journal

| last = Essam

| first = John

|author2=K. De'Bell |author3=J. Adler |author3-link= Joan Adler |author4=F. M. Bhatti

| title = Analysis of extended series for bond percolation on the directed square lattice

| journal = Physical Review B

| volume = 33

| issue = 2

| year = 1986

| pages = 1982–1986

| doi = 10.1103/PhysRevB.33.1982| pmid = 9938508

|bibcode = 1986PhRvB..33.1982E }} 0.644701(1),{{cite journal

| last = Baxter

| first = R. J.

|author2=A. J. Guttmann

| title = Series expansion of the percolation probability for the directed square lattice

| journal = J. Phys. A

| volume = 21

| issue = 15

| year = 1988

| pages = 3193–3204

| doi = 10.1088/0305-4470/21/15/008|bibcode = 1988JPhA...21.3193B }} 0.644701(1), 0.6447006(10), 0.64470015(5),{{cite journal

| last = Jensen

| first = Iwan

| title = Low-density series expansions for directed percolation on square and triangular lattices

| journal = J. Phys. A

| volume = 29

| issue = 22

| year = 1996

| pages = 7013–7040

| doi = 10.1088/0305-4470/29/22/007|bibcode = 1996JPhA...29.7013J | s2cid = 121332666

}} 0.644700185(5), 0.6447001(2), 0.643(2)

(1+1)-d triangular

| 3

| 0.595646(3), 0.5956468(5), 0.5956470(3)

| 0.478018(2), 0.478025(1), 0.4780250(4) 0.479(3)

(2+1)-d simple cubic, diagonal planes

| 3

| 0.43531(1), 0.43531411(10)

| 0.382223(7), 0.38222462(6) 0.383(3)

(2+1)-d square nn (= bcc)

| 4

| 0.3445736(3),{{cite journal

| last = Grassberger

| first = P.

| title = Local persistence in directed percolation

| journal = Journal of Statistical Mechanics: Theory and Experiment

| volume = 2009

| issue = 8

| year = 2009

| pages = P08021

| doi = 10.1088/1742-5468/2009/08/P08021 |bibcode = 2009JSMTE..08..021G |arxiv = 0907.4021 | s2cid = 119236556

}} 0.344575(15) 0.3445740(2)

| 0.2873383(1),{{cite journal

| last = Perlsman

| first = E.

|author2=S. Havlin

| title = Method to estimate critical exponents using numerical studies

| journal = Europhys. Lett.

| volume = 58

| issue = 2

| year = 2002

| pages = 176–181

| url =https://semanticscholar.org/paper/502191a5395e90141d8f22da5576825a693cda5a

| doi = 10.1209/epl/i2002-00621-7|bibcode = 2002EL.....58..176P | s2cid = 67818664

}}

0.287338(3){{cite journal

| last = Grassberger

| first = P.

|author2=Y.-C. Zhang

| title = "Self-organized" formulation of standard percolation phenomena

| journal = Physica A

| volume = 224

| issue = 1

| year = 1996

| pages = 169–179

| doi = 10.1016/0378-4371(95)00321-5|bibcode = 1996PhyA..224..169G }} 0.28733838(4) 0.287(3)

(2+1)-d fcc

| 0.199(2))

(3+1)-d hypercubic, diagonal

| 4

| 0.3025(10),{{cite journal

| last = Adler

| first = Joan

| author-link = Joan Adler

| author2 = J. Berger |author3=M. A. M. S. Duarte |author4=Y. Meir

| title = Directed percolation in 3+1 dimensions

| journal = Physical Review B

| volume = 37

| issue = 13

| year = 1988

| pages = 7529–7533

| doi = 10.1103/PhysRevB.37.7529| pmid = 9944046

|bibcode = 1988PhRvB..37.7529A }}

0.30339538(5)

| 0.26835628(5), 0.2682(2)

(3+1)-d cubic, nn

| 6

|0.2081040(4)

| 0.1774970(5)

(3+1)-d bcc

| 8

|0.160950(30), 0.16096128(3)

|0.13237417(2)

(4+1)-d hypercubic, diagonal

| 5

| 0.23104686(3)

| 0.20791816(2), 0.2085(2){{cite journal

| last = Blease

| first = J.

| title = Series expansions for the directed-bond percolation problem

| journal = J. Phys. C: Solid State Phys.

| volume = 10

| issue = 7

| year = 1977

| pages = 917–924

| doi = 10.1088/0022-3719/10/7/003| bibcode = 1977JPhC...10..917B}}

(4+1)-d hypercubic, nn

| 8

| 0.1461593(2), 0.1461582(3){{cite journal

| last = Grassberger

| first = Peter

| title = Logarithmic corrections in (4 + 1)-dimensional directed percolation

| journal = Physical Review E

| volume = 79

| issue = 5

| year = 2009

| pages = 052104

| doi = 10.1103/PhysRevE.79.052104| pmid = 19518501

|bibcode = 2009PhRvE..79e2104G |arxiv = 0904.0804 | s2cid = 23876626

}}

| 0.1288557(5)

(4+1)-d bcc

| 16

| 0.075582(17),{{cite journal

| last = Lübeck

| first = S.

|author2=R. D. Willmann

| title = Universal scaling behavior of directed percolation around the upper critical dimension

| journal = J. Stat. Phys.

| volume = 115

| issue = 5–6

| year = 2004

| pages = 1231–1250

| doi = 10.1023/B:JOSS.0000028059.24904.3b|arxiv = cond-mat/0401395 |bibcode = 2004JSP...115.1231L | citeseerx = 10.1.1.310.8700

| s2cid = 16267627

}}

0.0755850(3), 0.07558515(1)

| 0.063763395(5)

(5+1)-d hypercubic, diagonal

| 6

| 0.18651358(2)

| 0.170615155(5), 0.1714(1)

(5+1)-d hypercubic, nn

| 10

| 0.1123373(2)

| 0.1016796(5)

(5+1)-d hypercubic bcc

| 32

|0.035967(23), 0.035972540(3)

| 0.0314566318(5)

(6+1)-d hypercubic, diagonal

| 7

| 0.15654718(1)

| 0.145089946(3), 0.1458

(6+1)-d hypercubic, nn

| 12

| 0.0913087(2)

| 0.0841997(14)

(6+1)-d hypercubic bcc

| 64

| 0.017333051(2)

| 0.01565938296(10)

(7+1)-d hypercubic, diagonal

| 8

| 0.135004176(10)

| 0.126387509(3), 0.1270(1)

(7+1)-d hypercubic,nn

| 14

| 0.07699336(7)

| 0.07195(5)

(7+1)-d bcc

| 128

| 0.008 432 989(2)

| 0.007 818 371 82(6)

nn = nearest neighbors. For a (d + 1)-dimensional hypercubic system, the hypercube is in d dimensions and the time direction points to the 2D nearest neighbors.

= Directed percolation with multiple neighbors =

File:DirectedPercolationNeighborhoods.png

(1+1)-d square with z NN, square lattice for z odd, tilted square lattice for z even

class="wikitable"
Lattice ! z ! width=30% \| Site percolation threshold ! width=30% \| Bond percolation threshold
(1+1)-d square \| 3 \| \| 0.4395(3),{{cite journal \| last = Soares \| first = Danyel J. B. \| author2 = José S Andrade Jr \| author3=Hans J. Herrmann \| title = Precise calculation of the threshold of various directed percolation models on a square lattice \| journal = J. Phys. A: Math. Gen. \| volume = 38 \| issue = 21 \| year = 2006 \| pages = L413–L415 \| doi = 10.1088/0305-4470/38/21/L06 \| arxiv = cond-mat/0503408 }}
(1+1)-d square \| 5 \| \| 0.2249(3)
(1+1)-d square \| 7 \| \| 0.1470(2)
(1+1)-d square \| 9 \| \| 0.1081(2)
(1+1)-d square \| 11 \| \| 0.0851(2)
(1+1)-d square \| 13 \| \| 0.0701(2)
(1+1)-d tilted sq \| 2 \| \| 0.6447(2)
(1+1)-d tilted sq \| 4 \| \| 0.3272(2)
(1+1)-d tilted sq \| 6 \| \| 0.2121(3)
(1+1)-d tilted sq \| 8 \| \| 0.1553(3)
(1+1)-d tilted sq \| 10 \| \| 0.1220(2)
(1+1)-d tilted sq \| 12 \| \| 0.0999(2)

class="wikitable"

Lattice

! z

! width=30% | Site percolation threshold

! width=30% | Bond percolation threshold

(1+1)-d square

| 3

| 0.4395(3),{{cite journal

| last = Soares

| first = Danyel J. B.

| author2 = José S Andrade Jr

| author3=Hans J. Herrmann

| title = Precise calculation of the threshold of various directed percolation models on a square lattice

| journal = J. Phys. A: Math. Gen.

| volume = 38

| issue = 21

| year = 2006

| pages = L413–L415

| doi = 10.1088/0305-4470/38/21/L06

| arxiv = cond-mat/0503408

}}

(1+1)-d square

| 5

| 0.2249(3)

(1+1)-d square

| 7

| 0.1470(2)

(1+1)-d square

| 9

| 0.1081(2)

(1+1)-d square

| 11

| 0.0851(2)

(1+1)-d square

| 13

| 0.0701(2)

(1+1)-d tilted sq

| 2

| 0.6447(2)

(1+1)-d tilted sq

| 4

| 0.3272(2)

(1+1)-d tilted sq

| 6

| 0.2121(3)

(1+1)-d tilted sq

| 8

| 0.1553(3)

(1+1)-d tilted sq

| 10

| 0.1220(2)

(1+1)-d tilted sq

| 12

| 0.0999(2)

For large z, p_c ~ 1/z

= Site-Bond Directed Percolation =

p_b = bond threshold

p_s = site threshold

Site-bond percolation is equivalent to having different probabilities of connections:

P_0 = probability that no sites are connected

P_2 = probability that exactly one descendant is connected to the upper vertex (two connected together)

P_3 = probability that both descendants are connected to the original vertex (all three connected together)

Formulas:

P_0 = (1-p_s) + p_s(1-p_b)^2

P_2 = p_s p_b (1-p_b)

P_3 = p_s p_b^2

P_0 + 2P_2 + P_3 = 1

class="wikitable"
Lattice ! z ! width=20% \| p_s ! width=20% \| p_b ! width=20% \| P_0 ! width=20% \| P_2 ! width=20% \| P_3
(1+1)-d square \| 3 \| 0.644701 \| 1 \| 0.126237 \| 0.229062 \| 0.415639
\| \| 0.7 \| 0.93585 \| 0.148376 \| 0.196529 \| 0.458567
\| \| 0.75 \| 0.88565 \| 0.169703 \| 0.166059 \| 0.498178
\| \| 0.8 \| 0.84135 \| 0.192304 \| 0.134616 \| 0.538464
\| \| 0.85 \| 0.80190 \| 0.216143 \| 0.102242 \| 0.579373
\| \| 0.9 \| 0.76645 \| 0.241215 \| 0.068981 \| 0.620825
\| \| 0.95 \| 0.73450 \| 0.267336 \| 0.034889 \| 0.662886
\| \| 1 \| 0.705489 \| 0.294511 \| 0 \| 0.705489

class="wikitable"

Lattice

! z

! width=20% | p_s

! width=20% | p_b

! width=20% | P_0

! width=20% | P_2

! width=20% | P_3

(1+1)-d square

| 3

| 0.644701

| 1

| 0.126237

| 0.229062

| 0.415639

| 0.7

| 0.93585

| 0.148376

| 0.196529

| 0.458567

| 0.75

| 0.88565

| 0.169703

| 0.166059

| 0.498178

| 0.8

| 0.84135

| 0.192304

| 0.134616

| 0.538464

| 0.85

| 0.80190

| 0.216143

| 0.102242

| 0.579373

| 0.9

| 0.76645

| 0.241215

| 0.068981

| 0.620825

| 0.95

| 0.73450

| 0.267336

| 0.034889

| 0.662886

| 1

| 0.705489

| 0.294511

| 0

| 0.705489

= Exact critical manifolds of inhomogeneous systems =

Inhomogeneous triangular lattice bond percolation

$1 - p_1 - p_2 - p_3 + p_1 p_2 p_3 = 0$

Inhomogeneous honeycomb lattice bond percolation = kagome lattice site percolation

$1 - p_1 p_2 - p_1 p_3 - p_2 p_3+ p_1 p_2 p_3 = 0$

Inhomogeneous (3,12^2) lattice, site percolation{{cite journal

| last = Wu

| first = F. Y.

| title = Critical frontier of the Potts and percolation models on triangular-type and kagome-type lattices I: Closed-form expressions

| journal = Physical Review E

| volume = 81

| issue = 6

| year = 2010

| pages = 061110

| doi = 10.1103/PhysRevE.81.061110 | pmid = 20866381

| arxiv = 0911.2514|bibcode = 2010PhRvE..81f1110W | s2cid = 31590247

}}

$1 - 3(s_1s_2)^2 + (s_1s_2)^3 = 0,$

or $s_1 s_2 = 1 - 2 \sin(\pi/18)$

Inhomogeneous union-jack lattice, site percolation with probabilities $p_1, p_2, p_3, p_4$ {{cite journal

| last = Damavandi

| first = Ojan Khatib

| author2 = Robert M. Ziff

| title = Percolation on hypergraphs with four-edges

| journal = J. Phys. A: Math. Theor.

| volume = 48

| issue = 40

| year = 2015

| pages = 405004

| doi = 10.1088/1751-8113/48/40/405004 | arxiv = 1506.06125|bibcode = 2015JPhA...48N5004K | s2cid = 118481075

}}

$p_3 = 1 - p_1; \qquad p_4 = 1 - p_2$

Inhomogeneous martini lattice, bond percolation

$1 - (p_1 p_2 r_3 + p_2 p_3 r_1 + p_1 p_3 r_2)
- (p_1 p_2 r_1 r_2 + p_1 p_3 r_1 r_3 + p_2 p_3 r_2 r_3)
+ p_1 p_2 p_3 ( r_1 r_2 + r_1 r_3 + r_2 r_3)
+ r_1 r_2 r_3 (p_1 p_2 + p_1 p_3 + p_2 p_3)
- 2 p_1 p_2 p_3 r_1 r_2 r_3 = 0$

Inhomogeneous martini lattice, site percolation. r = site in the star

$1 - r (p_1 p_2 + p_1 p_3 + p_2 p_3 - p_1 p_2 p_3) = 0$

Inhomogeneous martini-A (3–7) lattice, bond percolation. Left side (top of "A" to bottom): $r_2,\ p_1$ . Right side: $r_1, \ p_2$ . Cross bond: $\ r_3$ .

$1 - p_1 r_2 - p_2 r_1 - p_1 p_2 r_3 - p_1 r_1 r_3
- p_2 r_2 r_3 + p_1 p_2 r_1 r_3 + p_1 p_2 r_2 r_3
+ p_1 r_1 r_2 r_3+ p_2 r_1 r_2 r_3 - p_1 p_2 r_1 r_2 r_3 = 0$

Inhomogeneous martini-B (3–5) lattice, bond percolation

Inhomogeneous martini lattice with outside enclosing triangle of bonds, probabilities $y, x, z$ from inside to outside, bond percolation{{cite journal

| last = Wu

| first = F. Y.

| title =New Critical Frontiers for the Potts and Percolation Models

| journal = Physical Review Letters

| volume = 96

| issue = 9

| year = 2006

| pages = 090602

| doi = 10.1103/PhysRevLett.96.090602|arxiv = cond-mat/0601150 |bibcode = 2006PhRvL..96i0602W

| pmid=16606250| citeseerx = 10.1.1.241.6346

| s2cid = 15182833

}}

$1 - 3 z + z^3-(1-z^2) [3 x^2 y (1 + y - y^2)(1 + z) + x^3 y^2 (3 - 2 y)(1 + 2 z) ] = 0$

Inhomogeneous checkerboard lattice, bond percolation{{cite journal

| last = Ziff

| first = R. M.

|author2=C. R. Scullard |author3=J. C. Wierman |author4=M. R. A. Sedlock

| title = The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices

| journal = Journal of Physics A

| volume = 45

| issue = 49

| year = 2012

| pages = 494005

| doi = 10.1088/1751-8113/45/49/494005 |arxiv = 1210.6609 |bibcode = 2012JPhA...45W4005Z | s2cid = 2121370

}}

$1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4)
+ p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4 = 0$

Inhomogeneous bow-tie lattice, bond percolation

$1 - (p_1 p_2 + p_1 p_3 + p_1 p_4 + p_2 p_3 + p_2 p_4 + p_3 p_4)
+ p_1 p_2 p_3 + p_1 p_2 p_4 + p_1 p_3 p_4 + p_2 p_3 p_4
- u(1 - p_1 p_2 - p_3 p_4 + p_1 p_2 p_3 p_4) = 0$

where $p_1, p_2, p_3, p_4$ are the four bonds around the square and $u$ is the diagonal bond connecting the vertex between bonds $p_4, p_1$ and $p_2, p_3$ .

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Category:Percolation theory

Category:Critical phenomena

Category:Random graphs