Existence of a non-averaging regime for the self-avoiding walk on a high-dimensional infinite percolation cluster

TL;DR

This paper investigates the difference between quenched and annealed connective constants for self-avoiding walks on high-dimensional percolation clusters, revealing a non-averaging regime where these constants differ.

Contribution

It demonstrates that in high dimensions, a non-averaging regime exists with strict inequality between quenched and annealed connective constants for certain percolation probabilities.

Findings

Existence of a non-averaging regime in high dimensions.

Strict inequality between quenched and annealed constants for p in (p_c, p_c^{(2)}).

Identification of a threshold p_c^{(2)} > p_c where this regime occurs.

Abstract

Let Z_N be the number of self-avoiding paths of length N starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on Z^d with parameter p>p_c(Z^d). The object of this paper is to study the connective constant of the dilute lattice \limsup_{N\to \infty} Z_N^{1/N}, which is a non-random quantity. We want to investigate if the inequality \limsup_{N\to \infty} (Z_N)^{1/N} \le \lim_{N\to \infty} E[Z_N]^{1/N} obtained with the Borel-Cantelli Lemma is strict or not. In other words, we want to know the the quenched and annealed versions of the connective constant are the same. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when d is sufficiently large there exists p^{(2)}_c>p_c such that the inequality is strict for p\in (p_c,p^{(2)}_c).