Self-avoiding walk on the hypercube

TL;DR

This paper analyzes self-avoiding walks on high-dimensional hypercubes, identifying asymptotic behaviors, conjecturing phase transitions, and deriving an expansion for the connective constant using lace expansion techniques.

Contribution

It introduces an $N$-dependent connective constant for self-avoiding walks on hypercubes, proves its asymptotic expansion, and discusses phase transitions at different walk lengths.

Findings

Asymptotic expansion of the connective constant with integer coefficients.

Identification of the dilute phase regime for walks with length less than $2^{N/2}$.

Conjectures on phase transitions when walk length exceeds $2^{N/2}$.

Abstract

We study the number $c_n^{(N)}$ of $n$-step self-avoiding walks on the $N$-dimensional hypercube, and identify an $N$-dependent \emph{connective constant} $\mu_N$ and amplitude $A_N$ such that $c_n^{(N)}$ is $O(\mu_N^n)$ for all $n$ and $N$, and is asymptotically $A_N \mu_N^n$ as long as $n\le 2^{pN}$ for any fixed $p< \frac 12$. We refer to the regime $n \ll 2^{N/2}$ as the \emph{dilute phase}. We discuss conjectures concerning different behaviours of $c_n^{(N)}$ when $n$ reaches and exceeds $2^{N/2}$, corresponding to a critical window and a dense phase. In addition, we prove that the connective constant has an asymptotic expansion to all orders in $N^{-1}$, with integer coefficients, and we compute the first five coefficients $\mu_N = N-1-N^{-1}-4N^{-2}-26N^{-3}+O(N^{-4})$. The proofs are based on generating function and Tauberian methods implemented via the lace expansion, for which an introductory account is provided.