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

TL;DR

This paper demonstrates that the measure on markings of high-dimensional tori induced by a random walk becomes indistinguishable from uniform at a specific threshold, and establishes a cutoff phenomenon for the mixing time of lamplighter chains on these graphs.

Contribution

It introduces a general criterion for the uniformity of the uncovered set measure and cutoff phenomena in lamplighter chains on various graph families.

Findings

Measure on markings approaches uniform at threshold 1/2 of the cover time.

Lamplighter chain exhibits cutoff in total variation at the same threshold.

Provides asymptotics for correlation decay in the uncovered set.

Abstract

We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, 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 $T=1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph $\mathbf {Z}_2\wr \mathbf {Z}_n^d$, $d\geq3$, has a cutoff with threshold $1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. 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.