Asymptotic Capacity of the Range of Random Walks on Free Products of Graphs

TL;DR

This paper establishes the existence and properties of the asymptotic capacity of the range of random walks on free products of graphs, including a central limit theorem and real-analytic dependence on measures.

Contribution

It proves the asymptotic capacity exists, is almost surely constant and positive, and varies analytically with certain probability measures.

Findings

Asymptotic capacity exists and is positive.

Capacity is almost surely constant.

Central limit theorem for the capacity.

Abstract

In this article we prove existence of the asymptotic capacity of the range of random walks on free products of graphs. In particular, we will show that the asymptotic capacity of the range is almost surely constant and strictly positive. Furthermore, we provide a central limit theorem for the capacity of the range and show that it varies real-analytically in terms of finitely supported probability measures of constant support.