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September 2013 Mean field conditions for coalescing random walks
Roberto Imbuzeiro Oliveira
Ann. Probab. 41(5): 3420-3461 (September 2013). DOI: 10.1214/12-AOP813

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 $\mathbf{E}[\mathsf{C}]\approx2\mathsf{m}(G)$ 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 $\approx\mathsf{m}(G)/\bigl({\matrix{k\\2}}\bigr)$.

Citation

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Roberto Imbuzeiro Oliveira. "Mean field conditions for coalescing random walks." Ann. Probab. 41 (5) 3420 - 3461, September 2013. https://doi.org/10.1214/12-AOP813

Information

Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1285.60094
MathSciNet: MR3127887
Digital Object Identifier: 10.1214/12-AOP813

Subjects:
Primary: 60K35
Secondary: 60J27

Keywords: Coalescing random walks , exponential approximation , hitting times , voter model

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 5 • September 2013
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