Annals of Probability

Uniformity of the uncovered set of random walk and cutoff for lamplighter chains

Jason Miller and Yuval Peres

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We show that the measure on markings of Znd, d ≥ 3, with elements of {0, 1} given by i.i.d. fair coin flips on the range $\mathcal{R}$ of a random walk X run until time T and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold T = ½Tcov(Znd). As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph Z2Znd, d ≥ 3, has a cutoff with threshold ½Tcov(Znd). We give a general criterion under which both of these results hold; other examples for which this applies include bounded degree expander families, the intersection of an infinite supercritical percolation cluster with an increasing family of balls, the hypercube and the Caley graph of the symmetric group generated by transpositions. The proof also yields precise asymptotics for the decay of correlation in the uncovered set.

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Ann. Probab., Volume 40, Number 2 (2012), 535-577.

First available in Project Euclid: 26 March 2012

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Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 37A25: Ergodicity, mixing, rates of mixing

Random walk uncovered set lamplighter walk mixing time cutoff


Miller, Jason; Peres, Yuval. Uniformity of the uncovered set of random walk and cutoff for lamplighter chains. Ann. Probab. 40 (2012), no. 2, 535--577. doi:10.1214/10-AOP624.

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