Asymptotic Expansion of passage probability for finite-range random walk on Free Groups

TL;DR

This paper derives an asymptotic expansion for the probability that a finite-range random walk on a free group is at a specific element after many steps, using advanced Tauberian methods and focusing on infinite trees with bounded valence.

Contribution

It provides a detailed asymptotic expansion for passage probabilities of finite-range random walks on free groups, extending previous results with new analytical techniques.

Findings

Asymptotic expansion to all orders for passage probabilities

Application of Tauberian theorems to group random walks

Results applicable to infinite trees with bounded valence

Abstract

Given a finite-range random walk on a finitely generated free group , what is the asymptotic behaviour, as the number of steps goes to infinity, of the sequence of probabilities that the random walk is at a given element of the group? In this article, we answer this question by providing an asymptotic expansion to any order for the sequence of probabilities that an irreducible finite-range random walk on an infinite tree with bounded valence is at a given vertex (See the Theorem 6.10). We will work under the assumption that each vertex of the tree has valence greater than or equal to three, and we will assume that the tree is endowed with a cofinite action of an automorphism group preserving the step distribution. The answer to the above question will appear as an immediate Corollary of the previously mentioned Theorem. This article is part of a triptych with [Che25a] and [Che25c]. It relies on Tauberian results from [Che25a], used throughout the text to derive asymptotic expansions. The careful study of the objects introduced in this text is done in [Che25c]