The Einstein relation for random walks on graphs

TL;DR

This paper explores the Einstein relation linking volume growth, resistance, and exit times of random walks on weighted graphs, establishing it under various conditions to understand diffusive behaviors.

Contribution

It proves the Einstein relation for random walks on weighted graphs under multiple conditions, including volume doubling and time comparison principles.

Findings

Established Einstein relation under volume doubling conditions

Extended the relation to various resistance and volume growth scenarios

Provided a framework for analyzing diffusive behavior of random walks

Abstract

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs.