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.
However, a construction of Ding and Peres @cite_10 shows that cannot hold in general if the underlying graph is not a tree, even if it has bounded degree. The message that we take from this is that the total variation mixing time is geometrically robust in general. One of the main aims of this article is to show that there is robustness amongst a wider class than just trees: indeed, holds if the graph has bounded degree and is roughly isometric to a tree. We will explain Ding and Peres' construction in more detail in Section .
@cite_10
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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. Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ|I|. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum degree Δ and λ< λc=(Δ-1)Δ-1 (Δ-2)Δ. This improves on the previously known general bound of λ ≤ (2 ‾ Δ-2). The new regime includes the interesting case of λ=1 (uniform weights) and Δ ≤ 5. The previous bound required Δ ≤ 4 for uniform approximate counting and there is evidence that for Δ ≥ 6 the problem is hard. Note that λc is the critical activity for uniqueness of the Gibbs measure on the regular tree of degree Δ, i.e., for λ ≤ λc the probability that the root is in the independent set is asymptotically independent of the configuration on the leaves far below. Indeed, our analysis is focused on establishing decay of correlations with distance in the above weighted distribution. We show that on any graph of maximum degree Δ correlations decay with distance at least as fast as they do on the regular tree of the same degree. This resolves an open conjecture in statistical physics. Our comparison of a general graph with the tree uses an algorithmic argument yielding the approximation scheme mentioned above. Also, by existing arguments, establishing decay of correlations for all graphs and λ<λc gives that the Glauber dynamics is rapidly mixing in this regime. However, the implication from decay of correlations to rapid mixing of the dynamics is only known to hold for graphs of subexponential growth, and hence, our result regarding the Glauber dynamics is limited to this class of graphs. In this note, we demonstrate an instance of bounded-degree graphs of size @math , for which the total variation mixing time for the random walk is decreased by a factor of @math if we multiply the edge-conductances by bounded factors in a certain way. 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.