Entropy of random walk range on uniformly transient and on uniformly recurrent graphs

TL;DR

This paper investigates how the entropy of the set of vertices visited by a simple random walk on bounded degree graphs scales with the number of steps, revealing linear growth in transient graphs and sublinear growth in recurrent graphs with subexponential volume growth.

Contribution

It establishes a clear relationship between graph recurrence/transience and the entropy growth rate of the random walk's visited set, answering a previously open question.

Findings

Entropy grows linearly in transient graphs.

Entropy grows sublinearly in recurrent graphs with subexponential volume growth.

Provides a characterization of entropy behavior based on graph properties.

Abstract

We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1).