Given a finite, connected graph @math , the lamplighter chain on @math is the lazy random walk @math on the associated lamplighter graph @math . The mixing time of the lamplighter chain on the torus @math is known to have a cutoff at a time asymptotic to the cover time of @math if @math , and to half the cover time if @math . We show that the mixing time of the lamplighter chain on @math has a cutoff at @math times the cover time of @math as @math , where @math is an explicit weakly decreasing map from @math onto @math . In particular, as @math varies, the threshold continuously interpolates between the known thresholds for @math and @math . Perhaps surprisingly, we find a phase transition (non-smoothness of @math ) at the point @math , where high dimensional behavior ( @math for all @math ) commences. Here @math is the effective resistance from @math to @math in @math .
By combining the results of @cite_7 and @cite_5 , it is observed in @cite_4 that @math has a threshold at @math . Thus, gives the best bounds, since @math attains the lower bound and @math attains the upper bound. In @cite_3 , it is shown that @math when @math and more generally that @math whenever @math is a sequence of graphs with @math satisfying certain uniform local transience assumptions. This prompted the question [Section 7] Miller-Peres of whether for each @math there exists a (natural) family of graphs @math such that @math as @math . In this work we give an affirmative answer to this question by analyzing the lamplighter chain on a thin @math D torus.
@cite_5 @cite_3 @cite_4 @cite_7
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Under a natural hypothesis, the cover time for a finite Markov chain can be approximated by its expectation, as the size of state space tends to infinity. This result is deduced from an abstract result concerning covering, an unstructured set by i.i.d. arbitrarily distributed random subsets. 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 @math 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 Z2 ≀ Znd, 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. Given a finite graph @math , a vertex of the lamplighter graph @math consists of a zero-one labeling of the vertices of @math , and a marked vertex of @math . For transitive @math we show that, up to constants, the relaxation time for simple random walk in @math is the maximal hitting time for simple random walk in @math , while the mixing time in total variation on @math is the expected cover time on @math . The mixing time in the uniform metric on @math admits a sharp threshold, and equals @math multiplied by the relaxation time on @math , up to a factor of @math . For @math , the lamplighter group over the discrete two dimensional torus, the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . For @math when @math , the relaxation time is of order @math , the total variation mixing time is of order @math , and the uniform mixing time is of order @math . In particular, these three quantities are of different orders of magnitude. Let τ n (x) denote the time of first visit of a point x on the lattice torus Z 2 n = 2 2 nZ 2 by the simple random walk. The size of the set of a, n-late points £ n (α) = x ∈ Z 2 n : Tn(x) ≥ α4 π (n log n) 2 is approximately n 2(1-α) , for α ∈ (0,1) [£ n (α) is empty if a > 1 and n is large enough]. These sets have interesting clustering and fractal properties: we show that for β e (0, 1), a disc of radius n β centered at nonrandom x typically contains about n 2β(1-α β2) points from £ n (α) (and is empty if β < √α), whereas choosing the center x of the disc uniformly in £ n (α) boosts the typical number of α, n-late points in it to n 2β(1-α) . We also estimate the typical number of pairs of a, n-late points within distance n β of each other; this typical number can be significantly smaller than the expected number of such pairs, calculated by Brummelhuis and Hilhorst [Phys. A 176 (1991) 387-408]. On the other hand, our results show that the number of ordered pairs of late points within distance n β of each other is larger than what one might predict by multiplying the total number of late points, by the number of late points in a disc of radius n β centered at a typical late point.