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 non-overlapping dyadic rectangles, each of area 1/ n , where a dyadic rectangle is any rectangle that can be written in the form [ a 2 − s , ( a + 1)2 − s ] × [ b 2 − t , ( b + 1)2 − t ] for a , b , s , t ∈ ℤ ⩾ 0. 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 Ω( n 1.38 ), improving upon the previously best lower bound of Ω( n log n ) coming from the diameter of the chain.