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 [16]. 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 2 Z0. 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(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least (n1.38), improving upon the previously best lower bound of (n log n) coming from the diameter of the chain.

Original languageEnglish
JournalCombinatorics, Probability and Computing
DOIs
Publication statusAccepted/In press - 1 Aug 2018

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Flip
Tiling
Markov processes
Markov chain
Polynomials
Relaxation time
Polynomial
Rectangle
Mixing Time
Relaxation Time
Perpendicular bisector
Open Problems
Complement
Valid
Lower bound
Upper bound
Imply
Unit

Keywords

  • Random dyadic tilings
  • Rapid mixing
  • Spectral gap

Cite this

Polynomial mixing of the edge-flip markov chain for unbiased dyadic tilings. / Cannon, Sarah; Levin, David A.; Stauffer, Alexandre.

In: Combinatorics, Probability and Computing, 01.08.2018.

Research output: Contribution to journalArticle

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