Site recurrence for coalescing random walk

TL;DR

This paper investigates the recurrence properties of coalescing random walks on various graphs, establishing site recurrence and occupation probability bounds using duality with the voter model.

Contribution

It introduces new results on site recurrence for coalescing random walks on bounded degree graphs and Galton-Watson trees with exponential tail offspring distributions, utilizing duality techniques.

Findings

Proves site recurrence for coalescing random walks on bounded degree graphs.

Establishes occupation probability bounds and a 0-1 law for the process.

Shows similar recurrence results for non-backtracking coalescing processes on trees.

Abstract

Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.