Asymptotic Entropy of Random Walks on Free Products

TL;DR

This paper investigates the asymptotic entropy of random walks on free products, establishing its existence, providing multiple formulas, and relating it to escape rates and volume growth.

Contribution

It introduces three distinct formulas for the asymptotic entropy of random walks on free products, connecting entropy with escape rates and volume growth.

Findings

Proves the existence of asymptotic entropy for these random walks.

Derives three equivalent formulas for the entropy.

Links entropy to the rate of escape and volume growth.

Abstract

Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.