Annals of Probability

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

Jason Miller and Yuval Peres

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 26 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1332772713

Digital Object Identifier
doi:10.1214/10-AOP624

Mathematical Reviews number (MathSciNet)
MR2952084

Zentralblatt MATH identifier
1251.60058

Subjects
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

Keywords
Random walk uncovered set lamplighter walk mixing time cutoff

Citation

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. https://projecteuclid.org/euclid.aop/1332772713


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