Average Entropy of the Ranges for Simple Random Walks on Discrete Groups

TL;DR

This paper investigates the entropy of the range of simple random walks on discrete groups, establishing conditions for when this entropy vanishes, which relates to the recurrence or transience of the walk.

Contribution

It introduces the concept of average entropy of the ranges for simple random walks and characterizes recurrence through entropy vanishing conditions.

Findings

Average entropy of the ranges exists for simple random walks.

Vanishing average entropy characterizes recurrence or specific transience behaviors.

Recurrence is equivalent to the vanishing of the average entropy of associated weighted digraphs.

Abstract

Inspired by Benjamini et al (Ann. Inst. H. Poincar\'{e} Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the ranges. Some equivalent conditions for the vanishing of the average entropy of the ranges are given. Particularly, the average entropy of the ranges vanishes if and only if the random walk is recurrent or escaping to negative infinity without left jump. In order to characterize the recurrence further, we study the average entropy of the weighted digraphs formed by the random walk. We show that the random walk is recurrent if and only if the average entropy of the weighted digraphs vanishes.