Range of Random Walks on Free Products

TL;DR

This paper studies transient random walks on free product graphs, establishing the existence and positivity of their asymptotic range, its real-analytic dependence on measures, and a related central limit theorem.

Contribution

It proves the existence, positivity, and real-analytic variation of the asymptotic range, along with a central limit theorem for these random walks.

Findings

Asymptotic range exists and is positive

Range varies real-analytically with probability measures

Central limit theorem for the range

Abstract

In this article we consider transient random walks on free products of graphs. We prove that the asymptotic range of these random walks exists and is strictly positive. In particular, we show that the range varies real-analytically in terms of probability measures of constant support. Moreover, we prove a central limit theorem associated with the range of the random walk.