Occupation measure of random walks and wired spanning forests in balls of Cayley graphs

TL;DR

This paper analyzes the occupation times of random walks and the structure of wired spanning forests in Cayley graphs, providing bounds on their expected sizes related to the radius of the balls.

Contribution

It establishes new bounds on occupation times and component sizes in Cayley graphs, extending understanding of random walks and spanning forests in these structures.

Findings

Expected occupation time in a ball is O(r^{5/2})

Expected component size in wired spanning forests is O(r^{11/2})

Results apply to general transient Cayley graphs

Abstract

We show that for finite-range, symmetric random walks on general transient Cayley graphs, the expected occupation time of any given ball of radius $r$ is $O(r^{5/2})$.. We also study the volume-growth property of the wired spanning forests on general Cayley graphs, showing that the expected number of vertices in the component of the identity inside any given ball of radius $r$ is $O(r^{11/2})$.