Log-Concave Polynomials IV: Exchange Properties, Tight Mixing Times, and Faster Sampling of Spanning Trees
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We prove tight mixing time bounds for natural random walks on bases of matroids, determinantal distributions, and more generally distributions associated with log-concave polynomials. For a matroid of rank k on a ground set of n elements, or more generally distributions associated with log-concave polynomials of homogeneous degree k on n variables, we show that the down-up random walk, started from an arbitrary point in the support, mixes in time O(klog k). Our bound has no dependence on n or the starting point, unlike the previous analyses [ALOV19, CGM19], and is tight up to constant factors. The main new ingredient is a property we call approximate exchange, a generalization of well-studied exchange properties for matroids and valuated matroids, which may be of independent interest. Additionally, we show how to leverage down-up random walks to approximately sample spanning trees in a graph with n edges in time O(nlog^2 n), improving on the almost-linear time algorithm by Schild [Sch18]. Our analysis works on weighted graphs too, and is the first to achieve nearly-linear running time.
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