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

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 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)] x [b2(-t) , (b + 1)2(-t)] for a, b, s, t is an element of Z(>= 0). The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doi ng 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.