Moduli of continuity of local times of random walks on graphs in terms of the resistance metric

TL;DR

This paper establishes universal concentration estimates and a modulus of continuity for local times of random walks on weighted graphs using the resistance metric, with applications to fractals and cover time analysis.

Contribution

It introduces new concentration bounds and continuity results for local times based on the resistance metric, applicable to fractals and graphs with volume growth conditions.

Findings

Universal concentration estimates for local times

Modulus of continuity for local times under volume growth conditions

Applications to scaling limits and cover time bounds on fractals

Abstract

In this article, universal concentration estimates are established for the local times of random walks on weighted graphs in terms of the resistance metric. As a particular application of these, a modulus of continuity for local times is provided in the case when the graphs in question satisfy a certain volume growth condition with respect to the resistance metric. Moreover, it is explained how these results can be applied to self-similar fractals, for which they are shown to be useful for deriving scaling limits for local times and asymptotic bounds for the cover time distribution.