## Annals of Probability

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

#### 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

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

Digital Object Identifier
doi:10.1214/10-AOP624

Mathematical Reviews number (MathSciNet)
MR2952084

Zentralblatt MATH identifier
1251.60058

#### 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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