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On the coalescence time of reversible random walks
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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.
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Meeting and coalescence times for random walks in the largest component of the Erd\H{o}s-R\'enyi random graph
Expected meeting, coalescence, and voter-consensus times on the Erdős–Rényi giant are Θ(n) throughout the fixed-supercritical, slightly-supercritical, and critical regimes.
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