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We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, withelements 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 becomesindistinguishable 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 thelamplighter graph $\mathbf {Z}_2\wr \mathbf {Z}_n^d$, $d\geq3$, has a cutoffwith threshold $1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. We give a generalcriterion under which both of these results hold; other examples for which thisapplies include bounded degree expander families, the intersection of aninfinite supercritical percolation cluster with an increasing family of balls,the hypercube and the Caley graph of the symmetric group generated bytranspositions. The proof also yields precise asymptotics for the decay ofcorrelation in the uncovered set.

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