We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.
Kozma @cite_13 showed that the upper bound on the @math -mixing time given in @cite_0 is not always correct up to constant factors. He then asked the general question is the mixing time a geometric property?'' and conjectured that the mixing time was robust (up to constant factors) under rough isometry for bounded degree graphs.
@cite_0 @cite_13
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Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. Let @math be the largest component of the Erdős–Renyi random graph @math . The mixing time of random walk on @math in the strictly supercritical regime, p = c n with fixed c > 1, was shown to have order log2n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p = (1 + e) n where λ = e3n is bounded, Nachmias and Peres proved that the mixing time on @math is of order n. However, it was unclear how to interpolate between these results, and estimate the mixing time as the giant component emerges from the critical window. Indeed, even the asymptotics of the diameter of @math in this regime were only recently obtained by Riordan and Wormald, as well as the present authors and Kim. In this paper, we show that for p = (1 + e) n with λ = e3n → ∞ and λ = o(n), the mixing time on @math is with high probability of order (n λ)log2λ. In addition, we show that this is the order of the largest mixing time over all components, both in the slightly supercritical and in the slightly subcritical regime [i.e., p = (1 − e) n with λ as above]. 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. We examine the spectral prole bound of Goel, Montenegro and Tet ali for the L 1 mixing time of continuous-time random walk in reversible settings. We nd that it is precise up to a log log factor, and that this log log factor cannot be improved.