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Abstract
We give the first polynomial upper bound on the mixing time of the edgeflip 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 nonoverlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2s, (a + 1)2s] × [b2t, (b + 1)2t] for a, b, s, t 2 Z0. The edgeflip 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 edgeflip 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 language  English 

Title of host publication  Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques  20th International Workshop, APPROX 2017 and 21st International Workshop, RANDOM 2017 
Subtitle of host publication  Volume 81 
Publisher  Schloss Dagstuhl LeibnizZentrum fur Informatik GmbH, Dagstuhl Publishing 
ISBN (Electronic)  9783959770446 
DOIs  
Publication status  Published  1 Aug 2017 
Event  20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017  Berkeley, USA United States Duration: 16 Aug 2017 → 18 Aug 2017 
Conference
Conference  20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017 and the 21st International Workshop on Randomization and Computation, RANDOM 2017 

Country/Territory  USA United States 
City  Berkeley 
Period  16/08/17 → 18/08/17 
Keywords
 Random dyadic tilings
 Rapid mixing
 Spectral gap
ASJC Scopus subject areas
 Software
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 1 Finished

Early Career Fellowship  Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures
Stauffer, A.
Engineering and Physical Sciences Research Council
1/04/16 → 30/09/22
Project: Research council