Unimodularity for multi-type Galton-Watson trees

TL;DR

This paper characterizes when a reversible measure for simple random walk exists on labeled multi-type Galton-Watson trees, providing explicit descriptions under certain conditions, thus advancing understanding of unimodularity in complex tree structures.

Contribution

It establishes necessary and sufficient conditions for the existence of a reversible measure on multi-type Galton-Watson trees and provides explicit descriptions when types are vertex labels.

Findings

Identifies conditions for reversibility of measures on multi-type Galton-Watson trees.

Provides explicit descriptions of such measures when types correspond to vertex labels.

Advances understanding of unimodularity in multi-type branching structures.

Abstract

Fix $n\in\mathbb{N}$. Let $\mathbf{T}_n$ be the set of rooted trees $(T,o)$ whose vertices are labeled by elements of $\{1,...,n\}$. Let $\nu$ be a strongly connected multi-type Galton-Watson measure. We give necessary and sufficient conditions for the existence of a measure $\mu$ that is reversible for simple random walk on $\mathbf{T}_n$ and has the property that given the labels of the root and its neighbors, the descendant subtrees rooted at the neighbors of the root are independent multi-type Galton-Watson trees with conditional offspring distributions that are the same as the conditional offspring distributions of $\nu$ when the types are $\nu$ are ordered pairs of elements of $[n]$. If the types of $\nu$ are given by the labels of vertices, then we give an explicit description of such $\mu$.