Percolation critical exponents

{{Short description|Mathematical parameter in percolation theory}}

In the context of the physical and mathematical theory of percolation, a percolation transition is characterized by a set of universal critical exponents, which describe the fractal properties of the percolating medium at large scales and sufficiently close to the transition. The exponents are universal in the sense that they only depend on the type of percolation model and on the space dimension. They are expected to not depend on microscopic details such as the lattice structure, or whether site or bond percolation is considered. This article deals with the critical exponents of random percolation.

Percolating systems have a parameter $p\,\!$ which controls the occupancy of sites or bonds in the system. At a critical value $p_c\,\!$ , the mean cluster size goes to infinity and the percolation transition takes place. As one approaches $p_c\,\!$ , various quantities either diverge or go to a constant value by a power law in $|p - p_c|\,\!$ , and the exponent of that power law is the critical exponent. While the exponent of that power law is generally the same on both sides of the threshold, the coefficient or "amplitude" is generally different, leading to a universal amplitude ratio.

Description

Thermodynamic or configurational systems near a critical point or a continuous phase transition become fractal, and the behavior of many quantities in such circumstances is described by universal critical exponents. Percolation theory is a particularly simple and fundamental model in statistical mechanics which has a critical point, and a great deal of work has been done in finding its critical exponents, both theoretically (limited to two dimensions) and numerically.

Critical exponents exist for a variety of observables, but most of them are linked to each other by exponent (or scaling) relations. Only a few of them are independent, and the choice of the fundamental exponents depends on the focus of the study at hand. One choice is the set $\{\sigma,\, \tau\}\,\!$ motivated by the cluster size distribution, another choice is $\{d_\text{f},\, \nu\}\,\!$ motivated by the structure of the infinite cluster. So-called correction exponents extend these sets, they refer to higher orders of the asymptotic expansion around the critical point.

Definitions of exponents

= Self-similarity at the percolation threshold =

Percolation clusters become self-similar precisely at the threshold density $p_c\,\!$ for sufficiently large length scales, entailing the following asymptotic power laws:

The fractal dimension $d_\text{f}\,\!$ relates how the mass of the incipient infinite cluster depends on the radius or another length measure, $M(L) \sim L^{d_\text{f}}\,\!$ at $p=p_c\,\!$ and for large probe sizes, $L\to\infty\,\!$ . Other notation: magnetic exponent $y_h = D - d_f\,\!$ and co-dimension $\Delta_\sigma = d - d_f\,\!$ .

The Fisher exponent $\tau\,\!$ characterizes the cluster-size distribution $n_s\,\!$ , which is often determined in computer simulations. The latter counts the number of clusters with a given size (volume) $s\,\!$ , normalized by the total volume (number of lattice sites). The distribution obeys a power law at the threshold, $n_s \sim s^{-\tau}\,\!$ asymptotically as $s\to\infty\,\!$ .

The probability for two sites separated by a distance $\vec r\,\!$ to belong to the same cluster decays as $g(\vec r)\sim |\vec r|^{-2(d-d_\text{f})}\,\!$ or $g(\vec r)\sim |\vec r|^{-d+(2-\eta)}\,\!$ for large distances, which introduces the anomalous dimension $\eta\,\!$ . Also, $\delta = (d + 2 - \eta)/(d - 2 + \eta)$ and $\eta = 2 - \gamma/\nu$ .

The exponent $\Omega\,\!$ is connected with the leading correction to scaling, which appears, e.g., in the asymptotic expansion of the cluster-size distribution,

$n_s \sim s^{-\tau}(1+\text{const} \times s^{-\Omega})\,\!$ for $s\to\infty\,\!$ . Also, $\omega = \Omega/(\sigma \nu) = \Omega d_f$ .

For quantities like the mean cluster size $S \sim a_0 |p - p_c|^{-\gamma} (1 + a_1 (p - p_c)^{\Delta_1} +\ldots )$ , the corrections are controlled by the exponent $\Delta_1 = \Omega\beta\delta = \omega \nu$ .

The minimum or chemical distance or shortest-path exponent $d_\mathrm{min}$ describes how the average minimum distance $\langle \ell \rangle$ relates to the Euclidean distance $r$ , namely $\langle \ell \rangle \sim r^{d_\mathrm{min}}$ Note, it is more appropriate and practical to measure average $r$ , < $r$ > for a given $\ell$ . The elastic backbone has the same fractal dimension as the shortest path. A related quantity is the spreading dimension $d_\ell$ , which describes the scaling of the mass M of a critical cluster within a chemical distance $\ell$ as $M \sim \ell^{d_\ell}$ , and is related to the fractal dimension $d_f$ of the cluster by $d_\ell = d_f/d_\mathrm{min}$ . The chemical distance can also be thought of as a time in an epidemic growth process, and one also defines $\nu_t$ where $d_\mathrm{min} = \nu_t/\nu = z$ , and $z$ is the dynamical exponent. One also writes $\nu_\parallel = \nu_t$ .

Also related to the minimum dimension is the simultaneous growth of two nearby clusters. The probability that the two clusters coalesce exactly in time $t$ scales as $p(t) \sim t^{-\lambda}$ with $\lambda = 1 + 5/(4 d_\mathrm{min})$ .{{cite journal

| title = Exact critical exponent for the shortest-path scaling function in percolation

| last = Ziff

| first = R. M.

| year = 1999

| pages = L457–L459

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

| volume = 32

| issue = 43

| doi = 10.1088/0305-4470/32/43/101| arxiv = cond-mat/9907305

| bibcode = 1999JPhA...32L.457Z

| s2cid = 1605985

}}

The dimension of the backbone, which is defined as the subset of cluster sites

carrying the current when a voltage difference is applied between two sites far apart, is $d_\text{b}$ (or $d_\text{BB}$ ). One also defines $\xi=d-d_\text{b}$ .

The fractal dimension of the random walk on an infinite incipient percolation cluster is given by $d_w$ .

The spectral dimension $\tilde d$ such that the average number of distinct sites visited in an $N$ -step random walk scales as $N^{\tilde d}$ .

= Critical behavior close to the percolation threshold =

The approach to the percolation threshold is governed by power laws again, which hold asymptotically close to $p_c\,\!$ :

The exponent $\nu\,\!$ describes the divergence of the correlation length $\xi\,\!$ as the percolation transition is approached, $\xi \sim |p-p_c|^{-\nu}\,\!$ . The infinite cluster becomes homogeneous at length scales beyond the correlation length; further, it is a measure for the linear extent of the largest finite cluster. Other notation: Thermal exponent $y_t = 1/\nu$ and dimension $\Delta_\epsilon = d - 1/\nu$ .

Off criticality, only finite clusters exist up to a largest cluster size $s_\max\,\!$ , and the cluster-size distribution is smoothly cut off by a rapidly decaying function, $n_s \sim s^{-\tau} f(s/s_\max)\,\!$ . The exponent $\sigma$ characterizes the divergence of the cutoff parameter, $s_\max \sim |p-p_c|^{-1/\sigma}\,\!$ . From the fractal relation we have $s_\max \sim \xi^{d_\text{f}}\,\!$ , yielding $\sigma = 1/\nu d_\text{f}\,\!$ .

The density of clusters (number of clusters per site) $n_c$ is continuous at the threshold but its third derivative goes to infinity as determined by the exponent $\alpha$ : $n_c \sim A + B (p - p_c) + C (p - p_c)^2 + D_\pm |p - p_c|^{2 - \alpha} + \cdots$ , where $D_\pm$ represents the coefficient above and below the transition point.

The strength or weight of the percolating cluster, $P$ or $P_\infty$ , is the probability that a site belongs to an infinite cluster. $P$ is zero below the transition and is non-analytic. Just above the transition, $P\sim (p-p_c)^\beta\,\!$ , defining the exponent $\beta\,\!$ . $\ P$ plays the role of an order parameter.

The divergence of the mean cluster size $S=\sum_s s^2 n_s/p_c \sim |p-p_c|^{-\gamma}\,\!$ introduces the exponent $\gamma\,\!$ .

The gap exponent Δ is defined as Δ = 1/(β+γ) = 1/σ and represents the "gap" in critical exponent values from one moment $M_n$ to the next $M_{n+1}$ for $n > 2$ .

The conductivity exponent $t = \nu t'$ describes how the electrical conductivity $C$ goes to zero in a conductor-insulator mixture, $C\sim (p-p_c)^t\,\!$ . Also, $t' = \zeta$ .

= Surface critical exponents =

The probability a point at a surface belongs to the percolating or infinite cluster for $p\ge p_c$ is $P_\mathrm{surf}\sim (p-p_c)^{\beta_\mathrm{surf}}\,\!$ .

The surface fractal dimension is given by $d_\mathrm{surf} = d - 1 -\beta_\mathrm{surf}/\nu$ .{{cite journal

| last = Stauffer

| first = D.

|author2= A. Aharony

| title = Density profile of the incipient infinite percolation cluster

| journal = International Journal of Modern Physics C

| volume = 10

| issue = 5

| year = 1999

| pages = 935–940

| doi = 10.1142/S0129183199000735| bibcode = 1999IJMPC..10..935S

}}

Correlations parallel and perpendicular to the surface decay as $g_\parallel(\vec r)\sim |\vec r|^{2-d-\eta_\parallel}\,\!$ and $g_\perp(\vec r)\sim |\vec r|^{2-d-\eta_\perp}\,\!$ .{{cite journal

| last = Binder

| first = K.

|author2= P. C. Hohenberg

| title = Phase Transitions and Static Spin Correlations in Ising Models with Free Surfaces

| journal = Physical Review B

| volume = 6

| issue = 9

| year = 1972

| pages = 3461–3487

| doi = 10.1103/PhysRevB.6.3461| bibcode = 1972PhRvB...6.3461B

| url = https://link.aps.org/doi/10.1103/PhysRevB.6.3461

}}

The mean size of finite clusters connected to a site in the surface is $\chi_1\sim|p-p_c|^{-\gamma_1}$ .{{cite thesis

|type=PhD Thesis

|last=De'Bell

|first=Keith

|date=1980

|title=Surface effects in percolation

|publisher=University of London

}}{{cite book

|last=Binder

|first=K.

|date=1981

|editor-last=Domb

|editor-first=C.

|editor2-last=Lebowitz

|editor2-first=J. L.

|title=Phase Transitions and Critical Phenomena, Volume 8

|publisher=Academic Press

|pages=1–144

|chapter=Critical Behaviour at Surfaces

|isbn=978-0122203084

}}

The mean number of surface sites connected to a site in the surface is $\chi_{1,1}\sim|p-p_c|^{-\gamma_{1,1}}$ .

Scaling relations

= Hyperscaling relations =

: $\tau = \frac{d}{d_\text{f}} + 1\,\!$

: $d_\text{f} = d - \frac{\beta}{\nu}\,\!$

: $\eta = 2 + d - 2 d_\text{f}\,\!$

= Relations based on <math>\{\sigma, \tau\}</math> =

: $\alpha = 2 - \frac{\tau - 1}{\sigma}\,\!$

: $\beta = \frac{\tau - 2}{\sigma}\,\!$

: $\gamma = \frac{3-\tau}{\sigma}\,\!$

: $\beta+\gamma = \frac{1}{\sigma} = \nu d_f\,\!$

: $\nu = \frac{\tau-1}{\sigma d} = \frac{2\beta+\gamma}{d}\,\!$

: $\delta = \frac{1}{\tau-2}\,\!$

= Relations based on <math>\{d_\text{f}, \nu\}</math> =

: $\alpha = 2 - \nu d\,\!$

: $\beta = \nu (d - d_\text{f})\,\!$

: $\gamma = \nu (2 d_\text{f} - d)\,\!$

: $\sigma = \frac{1}{ \nu d_\text{f}}\,\!$

= Conductivity scaling relations =

: $d_w = d + \frac{t-\beta}{ \nu}\,\!$

: $t' = d_w - d_f\,\!$

: $\tilde d = 2 d_f/d_w$

= Surface scaling relations =

: $\eta_\parallel = 2 - d + 2 \beta_\mathrm{surf}/\nu\,\!$

: $d_\mathrm{surf} = d - 1 -\beta_\mathrm{surf}/\nu\,\!$

: $\eta_\parallel = 2\eta_\perp - \eta \,$ {{cite journal

| last = Lubensky

| first = T. C.

|author2= M. H. Rubin

| title = Critical phenomena in semi-infinite systems. I. $\epsilon$ expansion for positive extrapolation length

| journal = Physical Review B

| volume = 11

| issue = 11

| year = 1975

| pages = 4533–4546

| doi = 10.1103/PhysRevB.11.4533| bibcode = 1975PhRvB..11.4533L

| url = https://link.aps.org/doi/10.1103/PhysRevB.11.4533

}}

: $\gamma_{1} = \nu (2-\eta_\perp)\,\!$

: $\gamma_{1,1} = \nu (d-1-2 \beta_\mathrm{surf}/\nu)=\nu(1-\eta_\parallel)\,\!$ {{cite journal

| last = Pleimling

| first = M

| title = Critical phenomena at perfect and non-perfect surfaces

| journal = Journal of Physics A: Mathematical and General

| volume = 37

| year = 2004

| issue = 19

| pages = R79–R115

| doi = 10.1088/0305-4470/37/19/R01

| arxiv = cond-mat/0402574

| s2cid = 15712212

}}

: $\gamma + \nu = 2\gamma_1 - \gamma_{1,1} \,\!$

: $x_1 = \beta_\mathrm{surf}/\nu\,\!$

Exponents for standard percolation

class="wikitable"
{{math\|`d`}} ! {{math\|1}}{{cite journal \| last = Reynolds \| first = P. J. \|author2=H. E. Stanley \|author3=W. Klein \| title = Ghost fields, pair connectedness, and scaling: exact results in one-dimensional percolation \| journal = Journal of Physics A: Mathematical and General \| volume = 10 \| issue = 11 \| year = 1977 \| pages = L203–L209 \| url = https://doi.org/10.1088/0305-4470/10/11/007\| doi = 10.1088/0305-4470/10/11/007 \| bibcode = 1977JPhA...10L.203R }} ! {{math\|2}} ! {{math\|3}} ! {{math\|4}} ! {{math\|5}} ! {{math\|6 – `ε`}}{{cite journal \| title = Percolation theory \| last = Essam \| first = J. W. \| year = 1980 \| pages = 833–912 \| journal = Rep. Prog. Phys. \| volume = 43 \| issue = 7 \| url = http://stacks.iop.org/0034-4885/43/i=7/a=001\|bibcode = 1980RPPh...43..833E \|doi = 10.1088/0034-4885/43/7/001\| s2cid = 250755965 }} {{cite journal \| last = Harris \| first = A. B. \|author2= T. C. Lubensky \|author3= W. K. Holcomb \|author4= C. Dasgupta \| title = Renormalization-group approach to percolation problems \| journal = Physical Review Letters \| volume = 35 \| issue = 6 \| year = 1975 \| pages = 327–330 \| doi = 10.1103/PhysRevLett.35.327\| bibcode = 1975PhRvL..35..327H\| url = https://repository.upenn.edu/cgi/viewcontent.cgi?article=1329&context=physics_papers }} {{cite journal \| last = Priest \| first = R. G. \|author2= T. C. Lubensky \| title = Critical properties of two tensor models with application to the percolation problem \| journal = Physical Review B \| volume = 13 \| issue = 9 \| year = 1976 \| pages = 4159–4171 \| doi = 10.1103/PhysRevB.13.4159\| bibcode = 1976PhRvB..13.4159P \| url = https://link.aps.org/doi/10.1103/PhysRevB.13.4159 }} {{refn\|group=note\|For higher-order terms in the $\varepsilon$ expansions, see.{{cite journal \| last = Alcantara Bonfim \| first = 0. F. \| author2 = J E Kirkham \| author3 = A J McKane \| title = Critical exponents for the percolation problem and the Yang-Lee edge singularity \| journal = J. Phys. A: Math. Gen. \| volume = 14 \| issue = 9 \| pages = 2391–2413 \| year = 1981 \| doi = 10.1088/0305-4470/14/9/034 \| bibcode =1981JPhA...14.2391D \| doi-access = free }}{{cite journal \| last = Borinsky \| first = M \| author2=J. A. Gracey \| author3=M. V. Kompaniets \| author4=O. Schnetz \| title = Five loop renormalization of phi^3 theory with applications to the Lee-Yang edge singularity and percolation theory \| journal = Physical Review D \| year = 2021 \| volume = 103 \| page = 116024 \| doi = 10.1103/PhysRevD.103.116024 \| arxiv = 2103.16224\| s2cid = 232417253 }} }} ! {{math\|6 +}}
{{math\|`α`}} \| 1 \| –2/3 \| -0.625(3) -0.64(4) \| -0.756(40) -0.75(2) \| -0.870(1) \| $-1 + \frac{\varepsilon}{7} - \frac{443}{2^2 3^2 7^3} \varepsilon^2$ \| -1
{{math\|`β`}} \| 0 \| 0.14(3) {{cite journal \| last = Sykes \| first = M. F. \|author2= M. Glen \|author3=D. S. Gaunt \| title =The percolation probability for the site problem on the triangular lattice \| journal = J. Phys. A: Math. Gen. \| volume = 7 \| issue = 9 \| year = 1974 \| pages = L105–L108 \| doi = 10.1088/0305-4470/7/9/002 \|bibcode = 1974JPhA....7L.105S}} 5/36 \| 0.39(2) 0.4181(8) 0.41(1) {{cite journal \| last = Sur \| first = A. \|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 = J. Stat. Phys. \| volume = 15 \| issue = 5 \| year = 1976 \| pages = 345–353 \| doi = 10.1007/BF01020338 \|bibcode= 1976JSP....15..345S \| s2cid = 38734613 }} 0.405(25),{{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 }} 0.4273 0.4053(5) 0.429(4) \| 0.52(3) 0.639(20) 0.657(9) 0.6590 0.658(1) \| 0.66(5) 0.835(5) 0.830(10) 0.8457 0.8454(2) \| $1 - \frac{\varepsilon}{7}-\frac{61}{2^2 3^2 7^3} \varepsilon^2$ \| 1
{{math\|`γ`}} \| 1 \| 43/18 \| 1.6 1.80(5) 1.66(7) 1.793(3) 1.805(20) {{cite journal \| last = Adler \| first = J. \| author-link = Joan Adler \| author2= Y. Meir \|author3=A. Aharony \|author4=A.B. Harris \| title = Series Study of Percolation Moments in General Dimension \| journal = Phys. Rev. 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 }} 1.8357 1.819(3) 1.78(3) \| 1.6(1) 1.48(8) 1.422(16) 1.4500 1.435(15) 1.430(6) \| 1.3(1) {{cite journal \| last = Kirkpatrick \| first = Scott \| title = Percolation phenomena in higher dimensions: Approach to the mean-field limit \| journal = Phys. Rev. Lett. \| volume = 36 \| year = 1976 \| issue = 2 \| pages = 69–72 \| doi = 10.1103/PhysRevLett.36.69\| bibcode = 1976PhRvL..36...69K }} 1.18(7) {{cite journal \| last = Gaunt \| first = D. S. \| author2 = H. Ruskin \| title = Bond percolation processes in d dimensions \| journal = J. Phys. A: Math. Gen. \| volume = 11 \| year = 1978 \| issue = 7 \| pages =1369–1380 \| doi = 10.1088/0305-4470/11/7/025\| bibcode = 1978JPhA...11.1369G }} 1.185(5) 1.1817 1.1792(7) \| $1 + \frac{\varepsilon}{7}+\frac{565}{2^2 3^2 7^3} \varepsilon^2$ \| 1
{{math\|`δ`}} \| $\infty$ \| 91/5, 18 \| 5.29(6) {{ref\|derived\|*}} 5.3 {{cite journal \| last = Nakanishi \| first = H \|author2= H. E. Stanley \| title = Scaling studies of percolation phenomena in systems of dimensionality of two to seven: Cluster numbers \| journal = Physical Review B \| volume = 22 \| issue = 5 \| year = 1980 \| pages = 2466–2488 \| doi = 10.1103/PhysRevB.22.2466\| bibcode = 1980PhRvB..22.2466N}} 5.16(4) \| 3.9 3.198(6) 3.175(8) \| 3.0 2.3952(12) \| $2 + \frac{2}{7}\varepsilon+\frac{565}{2\cdot 3^2 7^3} \varepsilon^2$ \| 2
{{math\|`η`}} \| 1 \| 5/24 \| -0.046(8) -0.059(9) -0.07(5) -0.0470 −0.03(1) \| -0.12(4) -0.0944(28) -0.0929(9) -0.0954 -0.084(4) \| -0.075(20) -0.0565 −0.0547(10) \| $-\frac{\varepsilon}{21}-\frac{206}{3^3 7^3} \varepsilon^2$ \| 0
{{math\|`ν`}} \| 1 \| 1.33(5) {{cite journal \| last = Levenshteĭn \| first = M. E. \|author2= B. I. Shklovskiĭ \|author3=M. S. Shur \|author4=A. L. Éfros \| title = The relation between the critical exponents of percolation theory \| journal = Zh. Eksp. Teor. Fiz. \| volume = 69 \| year = 1975 \| pages = 386–392 \| bibcode = 1975JETP...42..197L }}, 4/3 \| 0.8(1), 0.80(5), 0.872(7) 0.875(1) 0.8765(18) 0.8960 0.8764(12){{cite journal \| last = Wang \| first = J. \|author2=Z. Zhou \|author3=W. Zhang \|author4=T. M. Garoni \|author5= Y. Deng \| title = Bond and site percolation in three dimensions \| journal = Physical Review E \| volume = 87 \| issue = 5 \| year = 2013 \| pages = 052107 \| doi = 10.1103/PhysRevE.87.052107 \|arxiv = 1302.0421 \|bibcode = 2013PhRvE..87e2107W \| pmid = 23767487\| s2cid = 14087496 }}, 0.8751(11) {{cite journal \| last = Hu \| first = H. \|author2=H. W. Blöte \|author3= R. M. Ziff \|author4= Y. Deng \| title = Short-range correlations in percolation at criticality \| journal = Physical Review E \| volume = 90 \| issue = 4 \| year = 2014 \| pages = 042106 \| arxiv = 1406.0130 \|bibcode = 2014PhRvE..90d2106H \| doi=10.1103/PhysRevE.90.042106\| pmid = 25375437 \| s2cid = 21410490 }} 0.8762(12) 0.8774(13){{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 }} 0.88(2) 0.8762(7) {{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}} \| 0.6782(50) 0.689(10){{cite journal \| last = Ballesteros \| first = H. G. \|author2=L. A. Fernández \|author3=V. Martín-Mayor \|author4=A. Muñoz Sudepe \|author5=G. Parisi \|author6= J. J. Ruiz-Lorenzo \| title = Measures of critical exponents in the four-dimensional site percolation \| journal = Physics Letters 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 }} 0.6920 0.693 0.6852(28) 0.6845(23) 0.6845(6){{Cite journal \|last1=Tan \|first1=Xiao-Jun \|last2=Deng \|first2=You-Jin \|last3=Jacobsen \|first3=Jesper Lykke \|date=2020 \|title=N-cluster correlations in four- and five-dimensional percolation \|url=https://link.springer.com/10.1007/s11467-020-0972-6 \|journal=Frontiers of Physics \|language=en \|volume=15 \|issue=4 \|page=41501 \|doi=10.1007/s11467-020-0972-6 \|issn=2095-0462\|arxiv=2006.10981 \|bibcode=2020FrPhy..1541501T \|s2cid=219956814 }} 0.686(2) 0.6842(16) \| 0.51(5) 0.569(5) cited in 0.571(3) 0.5746 {{cite journal \| last = Gracey \| first = J. A. \| title = Four loop renormalization of φ^3 theory in six dimensions \| journal = Phys. Rev. D \| volume = 92 \| issue = 2 \| year = 2015 \| pages = 025012 \| doi =10.1103/PhysRevD.92.025012 \| arxiv = 1506.03357\| bibcode = 2015PhRvD..92b5012G\| s2cid = 119205590 }} 0.5723(18) 0.5737(33) {{cite journal \| last = Zhang \| first = Zhongjin \| author2=Pengcheng Hou \| author3=Sheng Fang \| author4=Hao Hu \| author5=Youjin Deng \| title = Critical exponents and universal excess cluster number of percolation in four and five dimensions \| journal = Physica A: Statistical Mechanics and Its Applications \| year = 2021 \| volume = 580 \| page = 126124 \| doi = 10.1016/j.physa.2021.126124 \| arxiv = 2004.11289\| bibcode = 2021PhyA..58026124Z \| s2cid = 216080833 }} 0.5757(7) 0.5739(1) 0.5720(43) \| $\frac12 + \frac{5}{84} \varepsilon + \frac{589}{2^2 3^3 7^3} \varepsilon^2$ \| 1/2
{{math\|`σ`}} \| 1 \| 36/91 \| 0.42(6) {{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 = J. Phys. A: Math. Gen. \| volume = 9 \| issue = 5 \| year = 1976 \| pages = L43–L46 \| doi = 10.1088/0305-4470/9/5/002 \|bibcode = 1976JPhA....9L..43S}} 0.445(10) {{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 = Phys. Rev. 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 }} 0.4522(8) 0.4524(6) 0.4419 0.452(7) \|0.476(5) 0.4742 0.4789(14) \| 0.496(4) 0.4933 0.49396(13) \| $\frac12 - \frac{1}{98} \varepsilon^2$ \| 1/2
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$d_w$ \| \| 2.8784(8) \| \| \| \| \|
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$\eta_{\parallel}$ (surf) \| \| 2/3 \| 1.02(12) 1.08(10) \| 1.37(13) \| 1.7(6) \| \|
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! {{math|2}}

! {{math|3}}

! {{math|4}}

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{{refn|group=note|For higher-order terms in the $\varepsilon$ expansions, see.{{cite journal

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! {{math|6 +}}

| 1

| –2/3

| -0.625(3)
-0.64(4)

| -0.756(40)
-0.75(2)

| -0.870(1)

| $-1 + \frac{\varepsilon}{7} - \frac{443}{2^2 3^2 7^3} \varepsilon^2$

| -1

| 0

| 0.14(3) {{cite journal

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5/36

| 0.39(2)
0.4181(8)
0.41(1) {{cite journal

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0.4273

0.4053(5)

0.429(4)

| 0.52(3)
0.639(20)
0.657(9)
0.6590

0.658(1)

| 0.66(5)
0.835(5)
0.830(10)
0.8457

0.8454(2)

| $1 - \frac{\varepsilon}{7}-\frac{61}{2^2 3^2 7^3} \varepsilon^2$

| 1

| 43/18

| 1.6
1.80(5)
1.66(7)
1.793(3)
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1.8357
1.819(3)

1.78(3)

| 1.6(1)
1.48(8)
1.422(16)
1.4500
1.435(15)

1.430(6)

| 1.3(1)

{{cite journal

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1.18(7)

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1.185(5)
1.1817

1.1792(7)

| $1 + \frac{\varepsilon}{7}+\frac{565}{2^2 3^2 7^3} \varepsilon^2$

| 1

| $\infty$

| 91/5, 18

| 5.29(6) {{ref|derived|*}}
5.3

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5.16(4)

| 3.9
3.198(6)

3.175(8)

| 3.0

2.3952(12)

| $2 + \frac{2}{7}\varepsilon+\frac{565}{2\cdot 3^2 7^3} \varepsilon^2$

| 2

| 1

| 5/24

| -0.046(8)
-0.059(9)
-0.07(5)
-0.0470

−0.03(1)

| -0.12(4)
-0.0944(28)
-0.0929(9)
-0.0954

-0.084(4)

| -0.075(20)
-0.0565

−0.0547(10)

| $-\frac{\varepsilon}{21}-\frac{206}{3^3 7^3} \varepsilon^2$

| 0

| 1

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4/3

| 0.8(1),
0.80(5),
0.872(7)
0.875(1)
0.8765(18)
0.8960
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0.8762(12)
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| 0.6782(50)
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0.6920

0.693

0.6852(28)

0.6845(23)

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0.686(2)

0.6842(16)

| 0.51(5)
0.569(5) cited in
0.571(3)
0.5746 {{cite journal

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0.5737(33) {{cite journal

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0.5757(7)

0.5739(1)

0.5720(43)

| $\frac12 + \frac{5}{84} \varepsilon + \frac{589}{2^2 3^3 7^3} \varepsilon^2$

| 1/2

| 1

| 36/91

| 0.42(6) {{cite journal

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0.4522(8)
0.4524(6)

0.4419

0.452(7)

|0.476(5)
0.4742

0.4789(14)

| 0.496(4)
0.4933

0.49396(13)

| $\frac12 - \frac{1}{98} \varepsilon^2$

| 1/2

| 2

| 187/91

| 2.186(2) {{cite journal

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2.1888
2.189(2)
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2.18906(8)
2.18909(5)

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2.1938(12)

| 2.26
2.313(3){{cite journal

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2.3127(6)
2.313(2)
2.3124
2.3142(5)

2.3150(8)

| 2.33
2.412(4)
2.4171
2.419(1)

2.4175(2)

| $\frac52 - \frac{\varepsilon}{14} + \frac{313}{2^3 3^2 7^3} \varepsilon^2$

| 5/2

d_\text{f}

| 1

| 91/48

| 2.523(4) {{ref|derived|*}}
2.530(4) {{ref|derived|*}}
2.5230(1) {{cite journal

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3.0446(7)

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3.528
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1.3756(3)

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For $d=2$ , $d_\mathrm{b} = 2-(z^2-1)/12$ where $z$ satisfies $\sqrt{3}z / 4 + \sin(2 \pi z/3) = 0$ near $z = 2.3$ .

Exponents for protected percolation

In protected percolation, bonds are removed one at a time only from the percolating cluster. Isolated clusters are no longer modified. Scaling relations: $\beta' = \beta/(1+\beta)$ , $\gamma' = \gamma/(1+\beta)$ , $\nu' = \nu/(1+\beta)$ , $\tau' = \tau$ where the primed quantities indicated protected percolation

class="wikitable"
{{math\|`d`}} ! {{math\|1}} ! {{math\|2}} ! {{math\|3}} ! {{math\|4}} ! {{math\|5}} ! {{math\|6 – `ε`}} ! {{math\|6 +}}
{{math\|`β'`}} \| \| 5/41 \| 0.288 71(15){{Cite journal\|last1=Fayfar\|first1=Sean\|last2=Bretaña\|first2=Alex\|last3=Montfrooij\|first3=Wouter\|date=2021-01-15\|title=Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems\|journal=Journal of Physics Communications\|volume=5\|issue=1\|pages=015008\|doi=10.1088/2399-6528/abd8e9\|arxiv=2008.08258\|bibcode=2021JPhCo...5a5008F \|issn=2399-6528\|doi-access=free}} \| \| \| \|
{{math\|`γ'`}} \| \| 86/41 \| 1.3066(19) \| \| \| \|
{{math\|`τ'`}} \| \| 187/91 \| 2.1659(21) \| \| \| \|

class="wikitable"

! {{math|1}}

! {{math|2}}

! {{math|3}}

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| 5/41

| 0.288 71(15){{Cite journal|last1=Fayfar|first1=Sean|last2=Bretaña|first2=Alex|last3=Montfrooij|first3=Wouter|date=2021-01-15|title=Protected percolation: a new universality class pertaining to heavily-doped quantum critical systems|journal=Journal of Physics Communications|volume=5|issue=1|pages=015008|doi=10.1088/2399-6528/abd8e9|arxiv=2008.08258|bibcode=2021JPhCo...5a5008F |issn=2399-6528|doi-access=free}}

| 86/41

| 1.3066(19)

| 187/91

| 2.1659(21)

Exponents for directed percolation

Directed percolation (DP) refers to percolation in which the fluid can flow only in one direction along bonds—such as only in the downward direction on a square lattice rotated by 45 degrees. This system is referred to as "1 + 1 dimensional DP" where the two dimensions are thought of as space and time.

$\nu_\perp$ and $\nu_\parallel$ are the transverse (perpendicular) and longitudinal (parallel) correlation length exponents, respectively. Also $\zeta = 1/z = \nu_\perp / \nu_\parallel$ . It satisfies the hyperscaling relation $d/z = \eta + 2 \delta$ .

Another convention has been used for the exponent $z$ , which here we call $z'$ , is defined through the relation $\langle R^2 \rangle \sim t^{z'}$ , so that $z'= \nu_\parallel/\nu_\perp = 2/z$ . It satisfies the hyperscaling relation $d z' = 2 \eta + 4 \delta$ .

$\delta$ is the exponent corresponding to the behavior of the survival probability as a function of time: $P(t) \sim t^{-\delta}$ .

$\eta$ (sometimes called $\mu$ ) is the exponent corresponding to the behavior of the average number of visited sites at time $t$ (averaged over all samples including ones that have stopped spreading): $N(t) \sim t^{-\eta}$ .

The d(space)+1(time) dimensional exponents are given below.

class="wikitable"
{{math\|`d+1`}} ! {{math\|1+1}} ! {{math\|2+1}} ! {{math\|3+1}} ! {{math\|4 – `ε`}} {{cite journal \| last1 = Janssen \| first1 = H. K. \| last2 = Täuber \| first2=U. C. \| year = 2005 \| title = The field theory approach to percolation processes \| journal = Annals of Physics \| volume = 315 \| issue = 1 \| pages = 147–192 \| arxiv = cond-mat/0409670 \| bibcode = 2005AnPhy.315..147J \| doi = 10.1016/j.aop.2004.09.011 \| s2cid = 19033621 }} ! {{math\|Mean Field}}
{{math\|`β`}} \| 0.276486(8) {{cite journal \| last = Jensen \| first = I. \| year = 1999 \| 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 = 48 \| pages = 5233–5249 \| arxiv = cond-mat/9906036 \| bibcode = 1999JPhA...32.5233J \| doi = 10.1088/0305-4470/32/28/304 \| s2cid = 2681356 }} 0.276 7(3) \| 0.5834(30) 0.580(4) {{cite journal \| last = Wang \| first = Junfeng \| author2=Zongzheng Zhou \| author3=Qingquan Liu \| author4=Timothy M. Garoni \| author5= Youjin Deng \| year = 2013 \| title = High-precision Monte Carlo study of directed percolation in (d + 1) dimensions \| journal = Phys. Rev. E \| volume = 88 \| issue = 4 \| pages = 042102 \| arxiv = 1201.3006 \| bibcode = 2013PhRvE..88d2102W \| doi = 10.1103/PhysRevE.88.042102 \| pmid = 24229111 \| s2cid = 43011467 }} \| 0.813(9) {{cite journal \| last = Jensen \| first = I. \| year = 1992 \| title = Critical behavior of the three-dimensional contact process \| journal = Phys. Rev. A \| volume = 45 \| issue = 2 \| pages = R563–R566 \| bibcode = 1992PhRvA..45..563J \| doi = 10.1103/PhysRevA.45.R563 \| pmid = 9907104 }} 0.818(4) 0.82205 \| $1 - \frac{\varepsilon}{6}$ \| 1
{{math\|`δ,α` }} \| 0.159464(6) 0.15944(2) \| 0.4505(1) {{cite journal \| last1 = Voigt \| first1 = C. A. \| last2=Ziff \| first2= R. M. \| year = 1997 \| title = Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model \| journal = Phys. Rev. E \| volume = 56 \| issue = 6 \| pages = R6241–R6244 \| doi = 10.1103/PhysRevE.56.R6241 \| arxiv = cond-mat/9710211 \| bibcode = 1997PhRvE..56.6241V \| s2cid = 118952705 }} 0.451(3) {{cite journal \| last = Grassberger \| first = P. \| author2= Y. Zhang \| year = 1996 \| title = 'Self-organized' formulation of standard percolation phenomena \| journal = Physica A \| volume = 224 \| issue = 1–2 \| pages = 169 \| doi = 10.1016/0378-4371(95)00321-5 \| bibcode = 1996PhyA..224..169G }} 0.4509(5) 0.4510(4) 0.460(6) {{cite journal \| last = Grassberger \| first = P. \| year = 1989 \| title = Directed percolation in 2+1 dimensions \| journal = J. Phys. A: Math. Gen. \| volume = 22 \| issue = 17 \| pages = 3673–3679 \| doi = 10.1088/0305-4470/22/17/032 \| bibcode = 1989JPhA...22.3673G }} \| 0.732(4) {{cite book \| last = Henkel \| first = M. \| author2 = H. Hinrichsen \| author3 = S. Lŭbeck \| date=2008 \| title = Non-equilibrium phase transitions, Vol. 1: Absorbing phase transitions \| publisher=Springer, Dordrecht }} 0.7398(10) 0.73717 \| $1 - \frac{\varepsilon}{4}$ \| 1
{{math\|`η,θ`}} \| 0.313686(8) 0.31370(5) \| 0.2303(4) {{cite journal \| last = Perlsman \| first = E. \| author2= S. Havlin \| year = 2002 \| title = Method to estimate critical exponents using numerical studies \| journal = EPL \| volume = 58 \| issue = 2 \| pages = 176–181 \| doi = 10.1209/epl/i2002-00621-7 \| bibcode = 2002EL.....58..176P \| s2cid = 67818664 \| url = https://semanticscholar.org/paper/502191a5395e90141d8f22da5576825a693cda5a }} 0.2307(2) 0.2295(10) 0.229(3) 0.214(8) \|0.1057(3) 0.114(4) 0.12084 \| \|
{{math\| $\nu_\parallel$ }} \| 1.733847(6) 1.733825(25) {{cite journal \| last = Jensen \| first = Iwan \| year = 1996 \| title = Low-density series expansions for directed percolation on square and triangular lattices \| journal = J. Phys. A: Math. Gen. \| volume = 29 \| issue = 22 \| pages = 7013–7040 \| doi = 10.1088/0305-4470/29/22/007 \| bibcode = 1996JPhA...29.7013J }} 1.7355(15) 1.73(2) {{cite journal \| last = Amaral \| first = L. A. N. \| author2 = A.-L. Barabási \| author3 = S. V. Buldyrev \| author4 = S. T. Harrington \| author5 = S. Havlin \| author6 = R. Sadr-Lahijany \| author7 = H. E. Stanley \| year = 1995 \| title = Avalanches and the directed percolation depinning model: Experiments, simulations, and theory \| journal = Phys. Rev. E \| volume = 51 \| issue = 5 \| pages = 4655–4673 \| doi = 10.1103/PhysRevE.51.4655 \| pmid = 9963178 \| arxiv = cond-mat/9412047 \| bibcode = 1995PhRvE..51.4655A \| s2cid = 9953616 }} \| 1.16(5) 1.287(2) 1.295(6) \|1.106(3) 1.11(1) 1.10571 {{cite journal \| last = Janssen \| first = H. K. \| year = 1981 \| title = On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state \| journal = Annals of Physics \| volume = 42 \| issue = 2 \| pages = 151–154 \| doi = 10.1007/BF01319549 \| bibcode = 1981ZPhyB..42..151J \| s2cid = 120819248 }} \| \|
{{math\| $\nu_\perp$ }} \| 1.096854(4) 1.096844(14) 1.0979(10) \| 0.7333(75) 0.729(1) \| 0.584(5) 0.582(2) 0.58360 \| $\frac12 + \frac{\varepsilon}{16}$ \| $\frac12$
{{math\| $z=2/z'$ }} \| 1.580745(10) 1.5807(2) \| 1.7660(16) 1.765(3) 1.766(2) 1.7665(2) 1.7666(10) \| 1.88746 1.8990(4) 1.901(5) \| $2 - \frac{\varepsilon}{12}$ \| 2
{{math\|`γ`}} \| 2.277730(5) = 41/18?, 2.278(2) {{cite journal \| last = Essam \| first = J. W. \| author2= A. J. Guttmann \| author3 = K. De'Bell \| year = 1988 \| title = On two-dimensional directed percolation \| journal = J. Phys. A \| volume = 21 \| issue =19 \| pages = 3815–3832 \| doi = 10.1088/0305-4470/21/19/018 \| bibcode =1988JPhA...21.3815E }} \| 1.595(18) \| 1.237(23) \| \| 1
{{math\|`τ`}} \| 2.112(5), {{cite journal \| last = Dhar \| first = Deepak \| author2= Mustansir Barma \| year = 1981 \| title = Monte Carlo simulationof directed percolationon a square lattice \| journal = J. Phys. C: Solid State Phys. \| volume = 14 \| issue = 1\| pages = Ll-L6 \| doi = 10.1088/0022-3719/14/1/001 \| bibcode = 1981JPhC...14L...1D }} 2.1077(13),{{cite journal \| last = Owczarek \| first = A. L. \| author2= A. Rechnitzer \| author3= R. Brak \| author4= A. J. Guttmann \| year = 1997 \| title = On the hulls of directed percolation clusters \| journal = J. Physics A: Math. Gen. \| volume = 30 \| issue = 19\| pages = 6679 \| doi = 10.1088/0305-4470/30/19/011 \| bibcode = 1997JPhA...30.6679O }} 2.10825(8) \| \| \| \|

class="wikitable"

! {{math|1+1}}

! {{math|2+1}}

! {{math|3+1}}

! {{math|4 – ε}}

{{cite journal

| last1 = Janssen | first1 = H. K.

| last2 = Täuber | first2=U. C.

| year = 2005

| title = The field theory approach to percolation processes

| journal = Annals of Physics

| volume = 315 | issue = 1 | pages = 147–192

| arxiv = cond-mat/0409670

| bibcode = 2005AnPhy.315..147J

| doi = 10.1016/j.aop.2004.09.011

| s2cid = 19033621

}}

! {{math|Mean Field}}

| 0.276486(8)

{{cite journal

| last = Jensen | first = I.

| year = 1999

| 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 = 48 | pages = 5233–5249

| arxiv = cond-mat/9906036

| bibcode = 1999JPhA...32.5233J

| doi = 10.1088/0305-4470/32/28/304

| s2cid = 2681356

}}
0.276 7(3)

| 0.5834(30)
0.580(4)

{{cite journal

| last = Wang | first = Junfeng

| author2=Zongzheng Zhou

| author3=Qingquan Liu

| author4=Timothy M. Garoni

| author5= Youjin Deng

| year = 2013

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

| journal = Phys. Rev. E

| volume = 88 | issue = 4 | pages = 042102

| arxiv = 1201.3006

| bibcode = 2013PhRvE..88d2102W

| doi = 10.1103/PhysRevE.88.042102

| pmid = 24229111

| s2cid = 43011467

}}

| 0.813(9)

{{cite journal

| last = Jensen | first = I.

| year = 1992

| title = Critical behavior of the three-dimensional contact process

| journal = Phys. Rev. A

| volume = 45 | issue = 2 | pages = R563–R566

| bibcode = 1992PhRvA..45..563J

| doi = 10.1103/PhysRevA.45.R563

| pmid = 9907104

}}
0.818(4)
0.82205

| $1 - \frac{\varepsilon}{6}$

| 1

| 0.159464(6)
0.15944(2)

| 0.4505(1)

{{cite journal

| last1 = Voigt | first1 = C. A.

| last2=Ziff | first2= R. M.

| year = 1997

| title = Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model

| journal = Phys. Rev. E

| volume = 56 | issue = 6 | pages = R6241–R6244

| doi = 10.1103/PhysRevE.56.R6241

| arxiv = cond-mat/9710211

| bibcode = 1997PhRvE..56.6241V

| s2cid = 118952705

}}
0.451(3)

{{cite journal

| last = Grassberger | first = P.

| author2= Y. Zhang

| year = 1996

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

| journal = Physica A

| volume = 224 | issue = 1–2 | pages = 169

| doi = 10.1016/0378-4371(95)00321-5

| bibcode = 1996PhyA..224..169G

}}
0.4509(5)
0.4510(4)

0.460(6)

{{cite journal

| last = Grassberger | first = P.

| year = 1989

| title = Directed percolation in 2+1 dimensions

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

| volume = 22 | issue = 17 | pages = 3673–3679

| doi = 10.1088/0305-4470/22/17/032

| bibcode = 1989JPhA...22.3673G

}}

| 0.732(4)

{{cite book

| last = Henkel | first = M.

| author2 = H. Hinrichsen

| author3 = S. Lŭbeck

| date=2008

| title = Non-equilibrium phase transitions, Vol. 1: Absorbing phase transitions

| publisher=Springer, Dordrecht

}}
0.7398(10)
0.73717

| $1 - \frac{\varepsilon}{4}$

| 1

| 0.313686(8)
0.31370(5)

| 0.2303(4)

{{cite journal

| last = Perlsman | first = E.

| author2= S. Havlin

| year = 2002

| title = Method to estimate critical exponents using numerical studies

| journal = EPL

| volume = 58 | issue = 2 | pages = 176–181

| doi = 10.1209/epl/i2002-00621-7

| bibcode = 2002EL.....58..176P

| s2cid = 67818664

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

}}
0.2307(2)
0.2295(10)

0.229(3)

0.214(8)

|0.1057(3)
0.114(4)
0.12084

| 1.733847(6)
1.733825(25)

{{cite journal

| last = Jensen | first = Iwan

| year = 1996

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

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

| volume = 29

| issue = 22

| pages = 7013–7040

| doi = 10.1088/0305-4470/29/22/007

| bibcode = 1996JPhA...29.7013J

}}

1.7355(15)

1.73(2)

{{cite journal

| last = Amaral | first = L. A. N.

| author2 = A.-L. Barabási

| author3 = S. V. Buldyrev

| author4 = S. T. Harrington

| author5 = S. Havlin

| author6 = R. Sadr-Lahijany

| author7 = H. E. Stanley

| year = 1995

| title = Avalanches and the directed percolation depinning model: Experiments, simulations, and theory

| journal = Phys. Rev. E

| volume = 51 | issue = 5

| pages = 4655–4673

| doi = 10.1103/PhysRevE.51.4655

| pmid = 9963178

| arxiv = cond-mat/9412047

| bibcode = 1995PhRvE..51.4655A

| s2cid = 9953616

}}

| 1.16(5)
1.287(2)
1.295(6)

|1.106(3)
1.11(1)
1.10571

{{cite journal

| last = Janssen | first = H. K.

| year = 1981

| title = On the nonequilibrium phase transition in reaction-diffusion systems with an absorbing stationary state

| journal = Annals of Physics

| volume = 42 | issue = 2 | pages = 151–154

| doi = 10.1007/BF01319549

| bibcode = 1981ZPhyB..42..151J

| s2cid = 120819248

}}

| 1.096854(4)

1.096844(14)
1.0979(10)

| 0.7333(75)
0.729(1)

| 0.584(5)
0.582(2)
0.58360

| $\frac12 + \frac{\varepsilon}{16}$

| $\frac12$

{{math|

z=2/z'

}}

| 1.580745(10)
1.5807(2)

| 1.7660(16)
1.765(3)
1.766(2)
1.7665(2)
1.7666(10)

| 1.88746
1.8990(4)
1.901(5)

| $2 - \frac{\varepsilon}{12}$

| 2

| 2.277730(5) = 41/18?,
2.278(2)

{{cite journal

| last = Essam | first = J. W.

| author2= A. J. Guttmann

| author3 = K. De'Bell

| year = 1988

| title = On two-dimensional directed percolation

| journal = J. Phys. A

| volume = 21 | issue =19 | pages = 3815–3832

| doi = 10.1088/0305-4470/21/19/018

| bibcode =1988JPhA...21.3815E

}}

| 1.595(18)

| 1.237(23)

| 1

| 2.112(5),

{{cite journal

| last = Dhar | first = Deepak

| author2= Mustansir Barma

| year = 1981

| title = Monte Carlo simulationof directed percolationon a square lattice

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

| volume = 14 | issue = 1| pages = Ll-L6

| doi = 10.1088/0022-3719/14/1/001

| bibcode = 1981JPhC...14L...1D

}}
2.1077(13),{{cite journal

| last = Owczarek | first = A. L.

| author2= A. Rechnitzer

| author3= R. Brak

| author4= A. J. Guttmann

| year = 1997

| title = On the hulls of directed percolation clusters

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

| volume = 30 | issue = 19| pages = 6679

| doi = 10.1088/0305-4470/30/19/011

| bibcode = 1997JPhA...30.6679O

}}
2.10825(8)

Scaling relations for directed percolation

$\beta = \frac{\tau-2}{\sigma}$

$\gamma = \frac{3-\tau}{\sigma}$

$\tau = 2 + \frac{2}{1+\gamma/\beta}$

$\tilde \tau = \nu_\parallel-\beta$

$\eta = \gamma/\nu_\parallel-1$

$d_\mathrm{DP} = 2 - \beta/\nu_\parallel$

{{cite journal

| last = Deng | first = Youjin

| author2= Robert M. Ziff

| year = 2022

| title = The elastic and directed percolation backbone

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

| volume = 55 | issue = 24| pages = 244002

| doi = 10.1088/1751-8121/ac6843

| bibcode = 2022JPhA...55x4002D

| arxiv = 1805.08201| s2cid = 73528075

}}

$d_{b,\mathrm{DP}} = 2 - 2\beta/\nu_\parallel$

$\Delta = \beta + \gamma$

$dz' = 2 \eta + 4 \delta$

$d/z = \eta + 2 \delta$

Exponents for dynamic percolation

For dynamic percolation (epidemic growth of ordinary percolation clusters), we have

$P(t) \sim L^{-\beta/\nu} \sim (t^{1/d_\mathrm{min}})^{-\beta/\nu} = t^{-\delta}$ , implying

$\delta = \frac{\beta}{\nu d_\mathrm{min}} = \frac{d - d_f}{d_\mathrm{min}}$

For $N(t)\sim t^\eta$ , consider $N(\le s) \sim s^{3-\tau} \sim R^{d_f(3-\tau)} \sim t^{d_f(3-\tau)/d_\mathrm{min}}$ , and taking the derivative with respect to $t$ yields $N(t)\sim t^{d_f(3-\tau)/d_\mathrm{min}-1}$ , implying

$\eta = \frac{d_f(3-\tau)}{d_\mathrm{min}}-1 = \frac{2 d_f - d}{d_\mathrm{min}}-1$

Also, $z = d_\mathrm{min}$

Using exponents above, we find

class="wikitable"
{{math\|`d:`}} ! {{math\|2}} ! {{math\|3}} ! {{math\|4}} ! {{math\|5}} ! {{math\|6 – `ε`}} ! {{math\|Mean Field}}
{{math\| $\delta$ }} \| 0.09212 \| 0.34681 \| 0.59556 \| 0.8127 \| \| 1
{{math\| $\eta,\mu$ }} \| 0.584466 \| 0.48725 \| 0.30233 \| 0.1314 \| \| 0

class="wikitable"

! {{math|2}}

! {{math|3}}

! {{math|4}}

! {{math|5}}

! {{math|6 – ε}}

! {{math|Mean Field}}

{{math|

\delta

}}

| 0.09212

| 0.34681

| 0.59556

| 0.8127

| 1

{{math|

\eta,\mu

}}

| 0.584466

| 0.48725

| 0.30233

| 0.1314

| 0

Notes

References

= Further reading =

{{citation

| last1 = Stauffer

| first1 = D.

| last2 = Aharony

| first2 = A.

| title = Introduction to Percolation Theory

| edition = 2nd

| publisher = CRC Press

| year = 1994

| isbn = 978-0-7484-0253-3}}

Category:Percolation theory

Category:Critical phenomena

Category:Random graphs

Category:Critical exponents (phase transitions)

Percolation critical exponents

Description

Definitions of exponents

= Self-similarity at the percolation threshold =

= Critical behavior close to the percolation threshold =

= Surface critical exponents =

Scaling relations

= Hyperscaling relations =

= Relations based on <math>\{\sigma, \tau\}</math> =

= Relations based on <math>\{d_\text{f}, \nu\}</math> =

= Conductivity scaling relations =

= Surface scaling relations =

Exponents for standard percolation

Exponents for protected percolation

Exponents for directed percolation

Exponents for dynamic percolation

See also

Notes

References

= Further reading =