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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 [14]. 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 [a2 ^{s} , (a + 1)2 ^{s} ] × [b2 ^{t} , (b + 1)2 ^{t} ] for a, b, s, t EZ≥ 0. 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(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 language  English 

Pages (fromto)  365387 
Number of pages  23 
Journal  Combinatorics, Probability and Computing 
Volume  28 
Issue number  3 
Early online date  31 Oct 2018 
DOIs  
Publication status  Published  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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 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