Mean field conditions for coalescing random walks

TL;DR

This paper establishes conditions under which the coalescence time of coalescing random walks on finite graphs exhibits mean field behavior, approximating twice the mean meeting time, and extends these results to various graph types and nonreversible walks.

Contribution

It provides new sufficient conditions for mean field behavior of coalescing random walks, solving an open problem for vertex-transitive graphs and generalizing previous results.

Findings

Mean coalescence time approximates 2 times the mean meeting time.

Mean field behavior occurs on vertex-transitive graphs with small mixing times.

First meeting time among k walkers is approximately mean divided by k choose 2.

Abstract

The main results in this paper are about the full coalescence time $\mathsf{C}$ of a system of coalescing random walks over a finite graph $G$. Letting $\mathsf{m}(G)$ denote the mean meeting time of two such walkers, we give sufficient conditions under which $\mathbf{E}[\mathsf{C}]\approx 2\mathsf{m}(G)$ and $\mathsf{C}/\mathsf{m}(G)$ has approximately the same law as in the "mean field" setting of a large complete graph. One of our theorems is that mean field behavior occurs over all vertex-transitive graphs whose mixing times are much smaller than $\mathsf{m}(G)$; this nearly solves an open problem of Aldous and Fill and also generalizes results of Cox for discrete tori in $d\geq2$ dimensions. Other results apply to nonreversible walks and also generalize previous theorems of Durrett and Cooper et al. Slight extensions of these results apply to voter model consensus times, which are related to coalescing random walks via duality. Our main proof ideas are a strengthening of the usual approximation of hitting times by exponential random variables, which give results for nonstationary initial states; and a new general set of conditions under which we can prove that the hitting time of a union of sets behaves like a minimum of independent exponentials. In particular, this will show that the first meeting time among $k$ random walkers has mean $\approx\mathsf{m}(G)/\bigl({\matrix{k 2}}\bigr)$.