文章基本信息

标题：Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings
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作者：Sarah Cannon ; David A. Levin ; Alexandre Stauffer 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：81
页码：34:1-34:21
DOI：10.4230/LIPIcs.APPROX-RANDOM.2017.34
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain.
关键词：Random dyadic tilings; spectral gap; rapid mixing