Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings

Sarah Cannon, David A. Levin, Alexandre Stauffer

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Abstract

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 [14]. 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 ] × [b2 -t , (b + 1)2 -t ] for a, b, s, t EZ≥ 0. 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 Ω(n 1.38 ), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

Original languageEnglish
Pages (from-to)365-387
Number of pages23
JournalCombinatorics, Probability and Computing
Volume28
Issue number3
Early online date31 Oct 2018
DOIs
Publication statusPublished - 1 May 2019

Keywords

  • Random dyadic tilings
  • Rapid mixing
  • Spectral gap

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Applied Mathematics
  • Statistics and Probability
  • Computational Theory and Mathematics

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