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.
Highlights
Over the last couple of decades, extensive efforts have been devoted to studying the behaviour of random walks on general graphs, work that has yielded, for instance, estimates for the corresponding heat kernel and mixing times in terms of quantities such as volume growth and electrical resistance, which do not depend on precise structural information
Since the first time that local times of a simple random walk on a graph are non-zero everywhere gives the cover time, we believe that our results will provide another tool for studying the latter; this is a point upon which we will expand later in the article
We describe the extension of this local time continuity result to the infinite graph setting
Summary
Over the last couple of decades, extensive efforts have been devoted to studying the behaviour of random walks on general graphs, work that has yielded, for instance, estimates for the corresponding heat kernel and mixing times in terms of quantities such as volume growth and electrical resistance, which do not depend on precise structural information (see, for example, [7, 9, 31, 36]). In studying cover times of random walks on graphs, continuity properties of local times in terms of the resistance metric have previously been considered. This has been achieved by applying a general estimate on the fluctuations of a function on Euclidean space known in the literature as Garsia’s lemma (after [22, Lemma 1], cf [23, Lemma 1.1]) This approach was first used in [24, Theorem 2] to deduce the continuity of local times of Markov processes on the real line. We show that if we have a sequence of graphs such that the associated random walks admit a diffusion scaling limit that has jointly continuous local times, and a suitable local time equicontinuity result holds, it is further possible to obtain convergence of rescaled local times. We discuss an application to the study of cover times of random walks on graphs
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