On the coalescence time of reversible random walks

TL;DR

This paper establishes sharp bounds on the expected coalescence time of reversible random walks on finite graphs, linking it to the largest hitting time, and introduces new inequalities for meeting times.

Contribution

It proves that the expected coalescence time is bounded by a constant multiple of the largest hitting time, solving a problem posed by Aldous and Fill.

Findings

Expected coalescence time is at most a constant times the largest hitting time.

Sharp bounds are provided for all vertex-transitive graphs.

New exponential inequality for meeting times of reversible Markov chains.

Abstract

Consider a system of coalescing random walks where each individual performs random walk over a finite graph G, or (more generally) evolves according to some reversible Markov chain generator Q. Let C be the first time at which all walkers have coalesced into a single cluster. C is closely related to the consensus time of the voter model for this G or Q. We prove that the expected value of C is at most a constant multiple of the largest hitting time of an element in the state space. This solves a problem posed by Aldous and Fill and gives sharp bounds in many examples, including all vertex-transitive graphs. We also obtain results on the expected time until only k>1 clusters remain. Our proof tools include a new exponential inequality for the meeting time of a reversible Markov chain and a deterministic trajectory, which we believe to be of independent interest.