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Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

Part of: Designs and configurations Discrete geometry Theory of computing Markov processes

Published online by Cambridge University Press:  31 October 2018

S. CANNON ,
D. A. LEVIN  and
A. STAUFFER
S. CANNON
Affiliation:
Computer Science Division, University of California Berkeley, Berkeley, CA 94720-1776, USA (e-mail: sarah.cannon@berkeley.edu)
D. A. LEVIN
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403-1222, USA (e-mail: dlevin@uoregon.edu)
A. STAUFFER
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK (e-mail: a.stauffer@bath.ac.uk)
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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 ∈ ℤ⩾ 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(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 Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68Q87: Probability in computer science (algorithm analysis, random structures, phase transitions, etc.) 05B45: Tessellation and tiling problems 52C20: Tilings in $2$ dimensions
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 3 , May 2019 , pp. 365 - 387
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Supported by a grant from the Simons Foundation (#361047 to Sarah Cannon) and the National Science Foundation Graduate Research Fellowship Program under grant DGE-1650044. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

‡

Supported by a Marie Curie Career Integration Grant PCIG13-GA-2013-618588 DSRELIS, and an EPSRC Early Career Fellowship.

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