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on Square-lattice self-avoiding walk connective constant ($C_{38}$)

The statement

Let $\mathbb{Z}^2$ denote the square lattice graph with vertex set $\mathbb{Z}^2$ and edges between nearest neighbors (Euclidean distance $1$ ).

A self-avoiding walk (SAW) on a graph $G=(V,E)$ is a walk that visits no vertex more than once. In particular, for $\ell=1,2,\dots$ and $v\in V$ , let $N(v,\ell)$ denote the number of SAWs in $G$ of length $\ell$ starting at $v$ . [SSSY2014-Nv-ell]

The connective constant (also called the SAW growth constant) of a graph $G$ is defined by

\mu(G)\ :=\ \sup_{v\in V}\ \limsup_{\ell\to\infty} N(v,\ell)^{1/\ell}.

[SSSY2014-def-cc]

For vertex-transitive graphs, the $\limsup$ in the definition above can be replaced by a true limit. [SSSY2014-rem-vtx-limit]

For the square lattice $G=\mathbb{Z}^2$ , let $c_n$ be the number of $n$ -step SAWs starting at the origin. Then the square-lattice SAW connective constant is

C_{38} := \mu_{\mathbb{Z}^2}\ :=\ \lim_{n\to\infty} c_n^{1/n}.

What gets accepted

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