PHASE TRANSITIONS IN RANDOM DYADIC TILINGS AND RECTANGULAR DISSECTIONS

We study rectangular dissections of an n x n lattice region into rectangles of area n, where n = 2(k) for an even integer k. We show there is a natural edge-flipping Markov chain that connects the state space. A similar edge-flipping chain is known to connect the state space when restricted to dyadic tilings, where each rectangle is required to have the form R = [s2(u); (s + 1)2(u)] x [t2(v); (t + 1)2(v)]; where s; t; u, and v are nonnegative integers. The mixing time of this Markov chain for general rectangular dissections remains open, while recent work by Cannon, Levin, and Stauffer [Proceedings of APPROX/RANDOM 2017, pp. 34: 1{34: 21] gave a polynomial upper bound on the mixing time when restricting to dyadic tilings. We consider a weighted version of these Markov chains where, given a parameter lambda > 0; we would like to generate each rectangular dissection (or dyadic tiling) sigma with probability proportional to lambda(vertical bar sigma vertical bar) where vertical bar sigma vertical bar is the total edge length. We show there is a phase transition in the dyadic setting: when lambda < 1; the edge-flipping chain mixes in time O (n2), and when lambda > 1; the mixing time is exp(Omega(n(2))). The behavior for general rectangular dissections is more subtle, and even establishing ergodicity of the chain requires a careful inductive argument. As in the dyadic case, we show that the edge-flipping Markov chain for rectangular dissections requires exponential time when lambda > 1. Surprisingly, the chain also requires exponential time when lambda < 1, which we show using a di ff erent argument. Simulations suggest that the chain converges quickly at the isolated point lambda = 1, but this case remains open.