Connective constant

The connective constant is defined as follows. Let ${\displaystyle c_{n}}$ 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 ${\displaystyle c_{n+m}\leq c_{n}c_{m}}$. Then by applying Fekete's lemma to the logarithm of the above relation, the limit ${\displaystyle \mu =\lim _{n\rightarrow \infty }c_{n}^{1/n}}$ can be shown to exist. This number ${\displaystyle \mu }$ is called the connective constant, and clearly depends on the particular lattice chosen for the walk since ${\displaystyle c_{n}}$ does. The value of ${\displaystyle \mu }$ is precisely known only for two lattices, see below. For other lattices, ${\displaystyle \mu }$ has only been approximated numerically. It is conjectured that ${\displaystyle c_{n}\approx \mu ^{n}n^{\gamma -1}}$ as n goes to infinity, where ${\displaystyle \mu }$ and ${\displaystyle A}$, the critical amplitude, depend on the lattice, and the exponent ${\displaystyle \gamma }$, which is believed to be universal and dependent on the dimension of the lattice, is conjectured to be ${\displaystyle \gamma =43/32}$.[3]

Lattice	Connective constant
Hexagonal	${\displaystyle {\sqrt {2+{\sqrt {2}}}}\simeq 1.85}$
Triangular	${\displaystyle 4.15079(4)}$
Square	${\displaystyle 2.63815853032790(3)}$
Kagomé	${\displaystyle 2.56062}$
Manhattan	${\displaystyle 1.733535(3)}$
L-lattice	${\displaystyle 1.5657(15)}$
${\displaystyle (3.12^{2})}$ lattice	${\displaystyle 1.7110412...}$
${\displaystyle (4.8^{2})}$ lattice	${\displaystyle 1.80883001(6)}$

These values are taken from the 1998 Jensen–Guttmann paper [4] and a more recent paper by Jacobsen, Scullard and Guttmann.[5] The connective constant of the ${\displaystyle (3.12^{2})}$ 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

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

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

In 2010, Hugo Duminil-Copin and Stanislav Smirnov published the first rigorous proof of the fact that ${\displaystyle \mu ={\sqrt {2+{\sqrt {2}}}}}$ for the hexagonal lattice.[2] This had been conjectured by Nienhuis in 1982 as part of a larger study of O(n) models using renormalization techniques.[6] 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.[7] 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 ${\displaystyle \gamma }$ between two mid-edges ${\displaystyle a}$ and ${\displaystyle b}$, we define ${\displaystyle \ell (\gamma )}$ to be the number of vertices visited and its winding ${\displaystyle W_{\gamma }(a,b)}$ as the total rotation of the direction in radians when ${\displaystyle \gamma }$ is traversed from ${\displaystyle a}$ to ${\displaystyle b}$. The aim of the proof is to show that the partition function

${\displaystyle Z(x)=\sum _{\gamma :a\to H}x^{\ell (\gamma )}=\sum _{n=0}^{\infty }c_{n}x^{n}}$

converges for ${\displaystyle x<x_{c}}$ and diverges for ${\displaystyle x>x_{c}}$ where the critical parameter is given by ${\displaystyle x_{c}=1/{\sqrt {2+{\sqrt {2}}}}}$. This immediately implies that ${\displaystyle \mu ={\sqrt {2+{\sqrt {2}}}}}$.

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

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

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

${\displaystyle (p-v)F(p)+(q-v)F(q)+(r-v)F(r)=0,}$

where ${\displaystyle p,q,r}$ are the mid-edges emanating from ${\displaystyle v}$. 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 ${\displaystyle S_{T,L}}$ with 2L cells forming the left hand side, T cells across, and upper and lower sides at an angle of ${\displaystyle \pm \pi /3}$. (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 ${\displaystyle S_{T,L}}$ are given by

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

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

${\displaystyle A_{T,L}^{x}:=\sum _{\gamma \in S_{T,L}:a\to \alpha \setminus \{a\}}x^{\ell (\gamma )},\quad B_{T,L}^{x}:=\sum _{\gamma \in S_{T,L}:a\to \beta }x^{\ell (\gamma )},\quad E_{T,L}^{x}:=\sum _{\gamma \in S_{T,L}:a\to \epsilon \cup {\bar {\epsilon }}}x^{\ell (\gamma )}.}$

By summing the identity

${\displaystyle (p-v)F(p)+(q-v)F(q)+(r-v)F(r)=0}$

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

${\displaystyle 1=\cos(3\pi /8)A_{T,L}^{x_{c}}+B_{T,L}^{x_{c}}+\cos(\pi /4)E_{T,L}^{x_{c}}}$

after another clever computation. Letting ${\displaystyle L\to \infty }$, we get a strip domain ${\displaystyle S_{T}}$ and partition functions

${\displaystyle A_{T}^{x}:=\sum _{\gamma \in S_{T}:a\to \alpha \setminus \{a\}}x^{\ell (\gamma )},\quad B_{T}^{x}:=\sum _{\gamma \in S_{T}:a\to \beta }x^{\ell (\gamma )},\quad E_{T}^{x}:=\sum _{\gamma \in S_{T}:a\to \epsilon \cup {\bar {\epsilon }}}x^{\ell (\gamma )}.}$

It was later shown that ${\displaystyle E_{T,L}^{x_{c}}=0}$, but we do not need this for the proof.[8] We are left with the relation

${\displaystyle 1=\cos(3\pi /8)A_{T,L}^{x_{c}}+B_{T,L}^{x_{c}}}$.

From here, we can derive the inequality

${\displaystyle A_{T+1}^{x_{c}}-A_{T}^{x_{c}}\leq x_{c}(B_{T+1}^{x_{c}})^{2}}$

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

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 ${\displaystyle T_{-I}<\cdots <T_{-1}}$ and ${\displaystyle T_{0}>\cdots >T_{j}}$. Note that we can bound

${\displaystyle B_{T}^{x}\leq (x/x_{c})^{T}B_{T}^{x_{c}}\leq (x/x_{c})^{T}}$

which implies ${\displaystyle \prod _{T>0}(1+B_{T}^{x})<\infty }$. Finally, it is possible to bound the partition function by the bridge partition functions

${\displaystyle Z(x)\leq \sum _{T_{-I}<\cdots <T_{-1},\;T_{0}>\cdots >T_{j}}2\left(\prod _{k=-I}^{j}B_{T_{k}}^{x}\right)=2\left(\prod _{T>0}(1+B_{T}^{x})\right)^{2}<\infty .}$

And so, we have that ${\displaystyle \mu ={\sqrt {2+{\sqrt {2}}}}}$ as desired.

Nienhuis argued in favor of Flory's prediction that the mean squared displacement of the self-avoiding random walk ${\displaystyle \langle |\gamma (n)|^{2}\rangle }$ satisfies the scaling relation ${\displaystyle \langle |\gamma (n)|^{2}\rangle ={\frac {1}{c_{n}}}\sum _{n\;\mathrm {step\;SAW} }|\gamma (n)|^{2}=n^{2\nu +o(1)}}$, with ${\displaystyle \nu =3/4}$.[2] The scaling exponent ${\displaystyle \nu }$ and the universal constant ${\displaystyle 11/32}$ could be computed if the self-avoiding walk possesses a conformally invariant scaling limit, conjectured to be a Schramm–Loewner evolution with ${\displaystyle \kappa =8/3}$.[9]

• Percolation threshold

• Weisstein, Eric W. "Self-Avoiding Walk Connective Constant". MathWorld.