Rate of Escape on Free Products

TL;DR

This paper investigates the escape rate of random walks on free products of sets, establishing the existence of the speed and deriving multiple formulas for it using three different techniques.

Contribution

It proves the existence of the escape rate for random walks on free products and provides three distinct methods to compute it, enhancing understanding of their asymptotic behavior.

Findings

Existence of the escape rate for the considered random walks

Derivation of three equivalent formulas for the escape rate

Application of multiple techniques to analyze random walk speed

Abstract

Suppose we are given the free product $V$ of a finite family of finite or countable sets $(V_i)_{i\in\mathcal{I}}$ and probability measures on each $V_i$, which govern random walks on it. We consider a transient random walk on the free product arising naturally from the random walks on the $V_i$. We prove the existence of the rate of escape with respect to the block length, that is, the speed, at which the random walk escapes to infinity, and furthermore we compute formulas for it. For this purpose, we present three different techniques providing three different, equivalent formulas.