connective constant

The connective constant is defined as follows. Let $c\_n$ denote the number of n-step self-avoiding walks starting from a fixed origin point in the lattice. Since every n + m step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that $c\_\{n+m\}\; \backslash leq\; c\_n\; c\_m$. Then by applying Fekete's lemma to the logarithm of the above relation, the limit $\backslash mu=\backslash lim\_\{n\; \backslash rightarrow\; \backslash infty\}\; c\_n^\{1/n\}$ can be shown to exist. This number $\backslash mu$ is called the connective constant, and clearly depends on the particular lattice chosen for the walk since $c\_n$ does. The value of $\backslash mu$ is precisely known only for two lattices, see below. For other lattices, $\backslash mu$ has only been approximated numerically. It is conjectured that $c\_n\; \backslash approx\; \backslash mu^n\; n^\{\backslash gamma\; -1\}$ as n goes to infinity, where $\backslash mu$ and $A$, the critical amplitude, depend on the lattice, and the exponent $\backslash gamma$, which is believed to be universal and dependent on the dimension of the lattice, is conjectured to be $\backslash gamma=43/32$.

{{cite journal

|last1=Vöge |first1=Markus

|last2=Guttmann |first2=Anthony J.

|year=2003

|title=On the number of hexagonal polyominoes

|journal=Theoretical Computer Science

|volume=307 |issue=2 |pages=433–453

|doi=10.1016/S0304-3975(03)00229-9

|doi-access=}}

These values are taken from the 1998 Jensen–Guttmann paper

{{cite journal

|last1=Jensen |first1=I.

|last2=Guttmann |first2=A. J.

|year=1998

|title=Self-avoiding walks, neighbor-avoiding walks and trails on semi-regular lattices

|journal=Journal of Physics A

|volume=31 |issue=40 |pages=8137–45

|url=http://www.ms.unimelb.edu.au/~tonyg/articles/polygons.pdf

|doi=10.1088/0305-4470/31/40/008

|bibcode=1998JPhA...31.8137J

}} and a more recent paper by Jacobsen, Scullard and Guttmann.Jesper Lykke Jacobsen, Christian R Scullard and Anthony J Guttmann, 2016 J. Phys. A: Math. Theor. 49 494004

The connective constant of the $(3.12^2)$ lattice, since each step on the hexagonal lattice corresponds to either two or three steps in it, can be expressed exactly as the largest real root of the polynomial

: $x^\{12\}\; -\; 4x^8\; -\; 8x^7\; -\; 4x^6\; +\; 2x^4\; +\; 8x^3\; +\; 12x^2\; +\; 8x\; +\; 2$

given the exact expression for the hexagonal lattice connective constant. More information about these lattices can be found in the percolation threshold article.

==Duminil-Copin–Smirnov proof==

In 2010, Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that $\backslash mu=\backslash sqrt\{2\; +\; \backslash sqrt\{2\}\}$ for the hexagonal lattice. This had been conjectured by Nienhuis in 1982 as part of a larger study of O(n) models using renormalization techniques.

{{cite journal

|last1=Nienhuis |first1=Bernard

|year=1982

|title=Exact critical point and critical exponents of O(n) models in two dimensions

|journal=Physical Review Letters

|volume=49 |issue=15 |pages=1062–1065

|bibcode=1982PhRvL..49.1062N

|doi=10.1103/PhysRevLett.49.1062

}} The rigorous proof of this fact came from a program of applying tools from complex analysis to discrete probabilistic models that has also produced impressive results about the Ising model among others.

{{cite book

|last1=Smirnov |first1=Stanislav

|year=2010

|chapter=Discrete Complex Analysis and Probability

|title=Proceedings of the International Congress of Mathematicians (Hyderabad, India) 2010

|pages=565–621

|arxiv=1009.6077

|bibcode=2010arXiv1009.6077S

}} The argument relies on the existence of a parafermionic observable that satisfies half of the discrete Cauchy–Riemann equations for the hexagonal lattice. We modify slightly the definition of a self-avoiding walk by having it start and end on mid-edges between vertices. Let H be the set of all mid-edges of the hexagonal lattice. For a self-avoiding walk $\backslash gamma$ between two mid-edges $a$ and $b$, we define $\backslash ell(\backslash gamma)$ to be the number of vertices visited and its winding $W\_\{\backslash gamma\}(a,b)$ as the total rotation of the direction in radians when $\backslash gamma$ is traversed from $a$ to $b$. The aim of the proof is to show that the partition function

: $Z(x)=\backslash sum\_\{\backslash gamma:\; a\backslash to\; H\}x^\{\backslash ell(\backslash gamma)\}=\backslash sum\_\{n=0\}^\{\backslash infty\}c\_n\; x^n$

converges for

Given a domain $\backslash Omega$ in the hexagonal lattice, a starting mid-edge $a$, and two parameters $x$ and $\backslash sigma$, we define the parafermionic observable

$F(z)=\backslash sum\_\{\backslash gamma\backslash subset\backslash Omega:a\backslash to\; z\}\; e^\{-i\backslash sigma\; W\_\{\backslash gamma\}(a,z)\}x^\{\backslash ell(\backslash gamma)\}.$

If $x=x\_c=1/\backslash sqrt\{2\; +\; \backslash sqrt\{2\}\}$ and $\backslash sigma=5/8$, then for any vertex $v$ in $\backslash Omega$, we have

: $(p-v)F(p)\; +\; (q-v)F(q)\; +\; (r-v)F(r)=0,$

where $p,q,r$ are the mid-edges emanating from $v$. This lemma establishes that the parafermionic observable is divergence-free. It has not been shown to be curl-free, but this would solve several open problems (see conjectures). The proof of this lemma is a clever computation that relies heavily on the geometry of the hexagonal lattice.

Next, we focus on a finite trapezoidal domain $S\_\{T,L\}$ with 2L cells forming the left hand side, T cells across, and upper and lower sides at an angle of $\backslash pm\; \backslash pi/3$. (Picture needed.) We embed the hexagonal lattice in the complex plane so that the edge lengths are 1 and the mid-edge in the center of the left hand side is positioned at −1/2. Then the vertices in $S\_\{T,L\}$ are given by

: $V(S\_\{T,L\})=\backslash \{\; z\backslash in\; V(\backslash mathbb\{H\})\; :\; 0\; \backslash leq\; Re(z)\backslash leq\; \backslash frac\{3T+1\}\{2\},\; \backslash ;\; |\backslash sqrt\{3\}Im(z)-Re(z)|\backslash leq\; 3L\backslash \}.$

We now define partition functions for self-avoiding walks starting at $a$ and ending on different parts of the boundary. Let $\backslash alpha$ denote the left hand boundary, $\backslash beta$ the right hand boundary, $\backslash epsilon$ the upper boundary, and $\backslash bar\{\backslash epsilon\}$ the lower boundary. Let

: $$

A_{T,L}^x:=\sum_{\gamma \in S_{T,L}:a\to \alpha\setminus\{a\}} x^{\ell(\gamma)},\quad

B_{T,L}^x:=\sum_{\gamma \in S_{T,L}:a\to \beta} x^{\ell(\gamma)}, \quad

E_{T,L}^x:=\sum_{\gamma \in S_{T,L}:a\to \epsilon \cup \bar{\epsilon}} x^{\ell(\gamma)}.

By summing the identity

: $(p-v)F(p)\; +\; (q-v)F(q)\; +\; (r-v)F(r)=0$

over all vertices in $V(S\_\{T,L\})$ and noting that the winding is fixed depending on which part of the boundary the path terminates at, we can arrive at the relation

: $1=\backslash cos(3\backslash pi/8)\; A\_\{T,L\}^\{x\_c\}\; +\; B\_\{T,L\}^\{x\_c\}\; +\; \backslash cos(\backslash pi/4)\; E\_\{T,L\}^\{x\_c\}$

after another clever computation. Letting $L\backslash to\backslash infty$, we get a strip domain $S\_T$ and partition functions

: $$

A_{T}^x:=\sum_{\gamma \in S_{T}:a\to \alpha\setminus\{a\}} x^{\ell(\gamma)},\quad

B_{T}^x:=\sum_{\gamma \in S_{T}:a\to \beta} x^{\ell(\gamma)}, \quad

E_{T}^x:=\sum_{\gamma \in S_{T}:a\to \epsilon \cup \bar{\epsilon}} x^{\ell(\gamma)}.

It was later shown that $E\_\{T,L\}^\{x\_c\}=0$, but we do not need this for the proof.

{{Cite journal

|last1=Smirnov |first1=Stanislav

|year=2014

|title=The critical fugacity for surface adsorption of SAW on the honeycomb lattice is $1+\backslash sqrt\{2\}$

|journal=Communications in Mathematical Physics

|volume=326 |issue=3 |pages=727–754

|arxiv=1109.0358

|bibcode=2014CMaPh.326..727B

|doi=10.1007/s00220-014-1896-1

|s2cid=54799238

}}

We are left with the relation

: $1=\backslash cos(3\backslash pi/8)\; A\_\{T,L\}^\{x\_c\}\; +\; B\_\{T,L\}^\{x\_c\}$.

From here, we can derive the inequality

: $A\_\{T+1\}^\{x\_c\}\; -\; A\_\{T\}^\{x\_c\}\; \backslash leq\; x\_c\; (B\_\{T+1\}^\{x\_c\})^2$

And arrive by induction at a strictly positive lower bound for $B\_\{T\}^\{x\_c\}$. Since $Z(x\_c)\backslash geq\backslash sum\_\{T>0\}B\_T^\{x\_c\}=\backslash infty$, we have established that $\backslash mu\backslash geq\backslash sqrt\{2+\backslash sqrt\{2\}\}$.

For the reverse inequality, for an arbitrary self avoiding walk on the honeycomb lattice, we perform a canonical decomposition due to Hammersley and Welsh of the walk into bridges of widths and $T\_0>\backslash cdots\; >\; T\_j$. Note that we can bound

: $B\_T^x\backslash leq\; (x/x\_c)^T\; B\_T^\{x\_c\}\backslash leq\; (x/x\_c)^T$

which implies .

Finally, it is possible to bound the partition function by the bridge partition functions

:

And so, we have that $\backslash mu=\backslash sqrt\{2+\backslash sqrt\{2\}\}=2\backslash cos\backslash frac\{\backslash pi\}\{8\}$ as desired.

Nienhuis argued in favor of Flory's prediction that the mean squared displacement of the self-avoiding random walk $\backslash langle\; |\backslash gamma(n)|^2\; \backslash rangle$ satisfies the scaling relation

$\backslash langle\; |\backslash gamma(n)|^2\; \backslash rangle=\backslash frac\{1\}\{c\_n\}\; \backslash sum\_\{n\backslash ;\backslash mathrm\{step\backslash ;\; SAW\}\}|\backslash gamma(n)|^2=n^\{2\backslash nu\; +o(1)\}$,

with $\backslash nu=3/4$.

The scaling exponent $\backslash nu$ and the universal constant $11/32$ could be computed if the self-avoiding walk possesses a conformally invariant scaling limit, conjectured to be a Schramm–Loewner evolution with $\backslash kappa=8/3$.

{{cite book

|last1=Lawler |first1=Gregory F.

|last2=Schramm |first2=Oded

|last3=Werner |first3=Wendelin

|year=2004

|chapter=On the scaling limit of planar self-avoiding walk

|editor-last1=Lapidus |editor-first1=Michel L.

|editor-last2=van Frankenhuijsen |editor-first2=Machiel

|title=Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2: Multifractals, Probability and Statistical Mechanics, Applications

|series=Proceedings of Symposia in Pure Mathematics

|volume=72 |pages=339–364

|arxiv=math/0204277

|bibcode=2002math......4277L

|doi=10.1090/pspum/072.2/2112127

|isbn=9780821836385

|mr=2112127

|s2cid=16710180

}}

• Percolation threshold

{{reflist}}

• {{MathWorld|urlname=Self-AvoidingWalkConnectiveConstant|title=Self-Avoiding Walk Connective Constant}}

Category:Discrete geometry