The dependency digraph for irreducible finite-range random walk in free groups

TL;DR

This paper investigates the analytic and geometric properties of dependency digraphs related to finite-range random walks on free groups, contributing to understanding their asymptotic behaviors.

Contribution

It introduces the concept of a flooded cavern tree to analyze the dependency digraphs of irreducible finite-range random walks on free groups.

Findings

Defined flooded cavern trees for detailed dependency analysis

Studied properties of dependency digraphs in free groups

Provided groundwork for asymptotic expansion of passage probabilities

Abstract

This article is one of a triptych composed with [Che25a] and [Che25b], that aims at proving an asymptotic expansion to any order of the passage probability of an irreducible equivariant finite range random walk on a tree. In this text we study the analytic and geometric properties of the objects introduced in [Che25b]. To do so we introduce the definition of a ''flooded cavern tree'' enabling a precise study of some dependency digraph associated with a given irreducible finite-range random walk on a free group.