Transience of Simple Random Walks With Linear Entropy Growth

TL;DR

This paper investigates how linear growth in entropy for simple random walks on infinite graphs implies transience, establishing a connection between entropy and walk behavior with a necessary condition example.

Contribution

It demonstrates that linear entropy growth guarantees transience of simple random walks on bounded degree graphs, introducing a new entropy-based criterion for transience.

Findings

Linear entropy growth implies transience

A necessary condition on initial vertex independence

An example demonstrating the necessity of the condition

Abstract

Using the technique of evolving sets, we explore the connection between entropy growth and transience for simple random walks on connected infinite graphs with bounded degree. In particular we show that for a simple random walk starting at a vertex $x_0$, if the entropy after $n$ steps, $E_n$ is at least $Cn$ where the $C$ is independent of $x_0$, then the random walk is transient. We also give an example which demonstrates that the condition of $C$ being independent of $x_0$ is necessary.