Heat kernel estimates for strongly recurrent random walk on random media

TL;DR

This paper provides general heat kernel estimates for simple random walks on infinite random graphs, linking volume and resistance bounds to spectral dimension, with applications to long-range percolation clusters.

Contribution

It generalizes previous results to arbitrary infinite random graphs, establishing heat kernel bounds based on volume and resistance, and applies these to long-range percolation clusters.

Findings

Spectral dimension of the random graph is derived.

Heat kernel estimates depend on volume and resistance bounds.

Application to random walk on long-range percolation cluster.

Abstract

We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in \cite[Section 1,2]{BJKS}, and in particular, imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long range percolation cluster.