In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time.
Miller and Peres @cite_35 studied the connection between cover times and the mixing times for the random walks on corresponding lamplighter graphs. In particular, they designed a procedure which allowed to compute the cover time up to @math for a family of graphs that satisfy some transient'' condition. Miller pointed out that this procedure should also allow to compute the supremum of Gaussian free field up to @math . However, it seems that their method could not be extended to the case for general trees --- at least not without further substantial ingredient.
@cite_35
2033953301
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.