Non-coincidence of Quenched and Annealed Connective Constants on the supercritical planar percolation cluster

TL;DR

This paper investigates the growth rate of self-avoiding paths on supercritical percolation clusters, revealing that in two dimensions, the quenched connective constant is almost surely strictly less than its annealed counterpart, highlighting disorder effects.

Contribution

It proves that the quenched connective constant on supercritical percolation clusters in 2D is almost surely strictly less than the annealed expectation, extending understanding of disorder in self-avoiding walks.

Findings

Quenched connective constant is non-random almost surely.

In 2D, growth rate of paths is slower than expected value.

Method applies broadly to 2D models with disorder.

Abstract

In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on $\bbZ^d$. More precisely, we count $Z_N$ the number of self-avoiding paths of length $N$ on the infinite cluster, starting from the origin (that we condition to be in the cluster). We are interested in estimating the upper growth rate of $Z_N$, $\limsup_{N\to \infty} Z_N^{1/N}$, that we call the connective constant of the dilute lattice. After proving that this connective constant is a.s.\ non-random, we focus on the two-dimensional case and show that for every percolation parameter $p\in (1/2,1)$, almost surely, $Z_N$ grows exponentially slower than its expected value. In other word we prove that $\limsup_{N\to \infty} (Z_N)^{1/N} <\lim_{N\to \infty} \bbE[Z_N]^{1/N}$ where expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walk on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on specifics of percolation on $\bbZ^2$, so that our result can be extended to a large family of two dimensional models including general self-avoiding walk in random environment.