Bounds for the annealed return probability on large finite percolation clusters

TL;DR

This paper derives bounds for the expected return probability of a delayed random walk on finite clusters in large percolation models, especially at criticality, revealing asymptotic behaviors related to cluster size distributions.

Contribution

It introduces new bounds for the annealed return probability on percolation clusters, utilizing Hamiltonian properties of graph products, with sharp results at critical Bernoulli percolation on trees.

Findings

Bounds are tight for critical Bernoulli bond percolation on homogeneous trees.

Expected return probability decays as t^{-3/4} for large times.

Heavy-tailed cluster size distributions influence return probability bounds.

Abstract

Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond percolation on the homogeneous tree this bound is sharp. The asymptotic type of the expected return probability for large times t in this case is of order of the 3/4'th power of 1/t.