Open Access
March 2012 Uniformity of the uncovered set of random walk and cutoff for lamplighter chains
Jason Miller, Yuval Peres
Ann. Probab. 40(2): 535-577 (March 2012). DOI: 10.1214/10-AOP624


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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Jason Miller. Yuval Peres. "Uniformity of the uncovered set of random walk and cutoff for lamplighter chains." Ann. Probab. 40 (2) 535 - 577, March 2012.


Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1251.60058
MathSciNet: MR2952084
Digital Object Identifier: 10.1214/10-AOP624

Primary: 37A25 , 60D05 , 60J10

Keywords: Cutoff , Lamplighter walk , mixing time , Random walk , Uncovered set

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 2 • March 2012
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