Genealogy in critical generations of a diffusive random walk in random environment on trees

TL;DR

This paper studies the genealogy of vertices visited by a biased random walk on a Galton-Watson tree, revealing that coalescence occurs either in the recent or remote past, depending on the walk's behavior.

Contribution

It introduces a novel analysis of the genealogy structure of a diffusive random walk on trees, highlighting the dual nature of coalescence times.

Findings

Coalescence occurs either in the recent or remote past.

The structure of the visited vertices' genealogy depends on the walk's bias.

The study provides insights into the temporal dynamics of random walks on trees.

Abstract

We consider the range $R^{(n)}$, the tree made up of visited vertices by a diffusive null-recurrent randomly biased walk $\mathbb{X}$ on a Galton-Watson tree $\mathbb{T}$ up to the $n$-th return time to its root and we consider the following genealogy problem: pick two vertices uniformly at random in a generation of order $n$ in the tree $R^{(n)}$. Where does the coalescence occur? it turns out that the coalescence happens either in the recent past or in the remote past.