Polynomial mixing of the edgeflip Markov chain for unbiased dyadic tilings
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. 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}] \times [b2^{t}, (b+1)2^{t}] for nonnegative integers a,b,s,t. 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 \Omega(n^{1.38}), improving upon the previously best lower bound of \Omega(n\log n) coming from the diameter of the chain.
 Publication:

arXiv eprints
 Pub Date:
 November 2016
 arXiv:
 arXiv:1611.03636
 Bibcode:
 2016arXiv161103636C
 Keywords:

 Mathematics  Probability;
 Computer Science  Discrete Mathematics;
 Mathematics  Combinatorics