# Coalescing random walk on unimodular graphs

## TL;DR

This paper proves that coalescing random walks on unimodular graphs visit each site infinitely often and that opinions in the voter model have infinite expected lifetime, with implications for finite graphs.

## Contribution

It establishes new recurrence properties of coalescing random walks on unimodular graphs and extends results to finite graphs with controlled degree distributions.

## Key findings

- Coalescing random walk visits each site infinitely often almost surely.
- Voter model opinions have infinite expected lifetime on such graphs.
- Results apply to both infinite unimodular and finite graphs with bounded degree distributions.

## Abstract

Coalescing random walk on a unimodular random rooted graph for which the root has finite expected degree visits each site infinitely often almost surely. A corollary is that an opinion in the voter model on such graphs has infinite expected lifetime. Additionally, we deduce an adaptation of our main theorem that holds uniformly for coalescing random walk on finite random unimodular graphs with degree distribution stochastically dominated by a probability measure with finite mean.