Transience in growing subgraphs via evolving sets

TL;DR

This paper extends evolving set techniques to analyze time-varying conductance models, providing tight heat kernel bounds and demonstrating the transience of certain random walks on Z^d with dynamic conductances.

Contribution

It introduces a novel application of evolving sets to time-varying conductance models and proves transience results for random walks with dynamic edge and vertex conductances.

Findings

Established tight heat kernel upper bounds for time-varying conductance models.

Proved transience of lazy random walks on Z^d with dynamic conductances.

Confirmed part of a conjecture from previous work on evolving sets.

Abstract

We extend the use of random evolving sets to time-varying conductance models and utilize it to provide tight heat kernel upper bounds. It yields the transience of any uniformly lazy random walk, on Z^d, d>=3, equipped with uniformly bounded above and below, independently time-varying edge conductances, of (effectively) non-decreasing in time vertex conductances (i.e. reversing measure), thereby affirming part of [ABGK, Conj. 7.1].