How quickly can we sample a uniform domino tiling of the $$2L\times 2L$$ square via Glauber dynamics?

Abstract

The prototypical problem we study here is the following. Given a \(2L\times 2L\) square, there are approximately \(\exp (4KL^2/\pi )\) ways to tile it with dominos, i.e. with horizontal or vertical \(2\times 1\) rectangles, where \(K\approx 0.916\) is Catalan’s constant (Kasteleyn in Physica 27:1209–1225, 1961; Temperley and Fisher in Phil Mag 6:1061–1063, 1997). A conceptually simple (even if computationally not the most efficient) way of sampling uniformly one among so many tilings is to introduce a Markov Chain algorithm (Glauber dynamics) where, with rate \(1\), two adjacent horizontal dominos are flipped to vertical dominos, or vice-versa. The unique invariant measure is the uniform one and a classical question (Levin et al. in Am Math Soc, 2009; Luby et al. SIAM J Comput 31:167–192, 2001; Wilson in Ann Appl Probab 14:274–325, 2004) is to estimate the time \(T_\mathrm{mix}\) it takes to approach equilibrium (i.e. the running time of the algorithm). Luby et al. in (SIAM J Comput 31:167–192, 2001) and Randall and Tetali in (J Math Phys 41(3): 1598–1614, 2000) fast mixing was proven: \(T_\mathrm{mix}=O(L^C)\) for some finite \(C\). Here, we go much beyond and show that \(c L^2\le T_\mathrm{mix}\le L^{2+o(1)}\). Our result applies to rather general domain shapes (not just the \(2L\times 2L\) square), provided that the typical height function associated to the tiling is macroscopically planar in the large \(L\) limit, under the uniform measure [this is the case for instance for the Temperley-type boundary conditions considered in Kenyon (Ann Probab 28:759–795, 2000)]. Also, our method extends to some other types of tilings of the plane, for instance the tilings associated to dimer coverings of the hexagon or square-hexagon lattices.