Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings

TL;DR

This paper establishes the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving a problem posed in 2002 and providing bounds on relaxation and mixing times.

Contribution

It proves a polynomial upper bound on the relaxation and mixing times of the edge-flip Markov chain for dyadic tilings, a problem open for two decades.

Findings

Relaxation time at most O(n^{4.09})

Mixing time at most O(n^{5.09})

Lower bound on relaxation time is (n^{1.38})

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. 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}] \times [b2^{-t}, (b+1)2^{-t}] for non-negative integers a,b,s,t. 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(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.