Asymptotics of random domino tilings of rectangular Aztec diamonds

Abstract

We consider asymptotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, the explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations of the height functions to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.

Random tilings with the GPU

“…Figure 13. A tiling of a rectangular Aztec diamond, with the Arctic curve computed by [BK16] superimposed in red. Figure 14.…”

Fourier transform on high-dimensional unitary groups with applications to random tilings

“…it borders one of the dominos), then h(v) = h(u) ± 1, where the sign is + is (u, v) has a dark square on the left and − otherwise. [BK,Lemma 3.11] explains how for simply connected domains Definition 4.7 is matched to the above local definition of the height function by an affine change of coordinates. Note, however, that for multiple connected domains the local definition has an inconvenient feature: when we loop around a hole, the height function will pick up a non-zero increment.…”

“…It is shown in [BK,Proposition 6.2] that the above equation has a non-real complex root in the upper half-plane U if and only if (x, y) belongs to the liquid region L left of the left rectangle. Moreover, the resulting map (x, y) → z(x, y) is a smooth bijection between L left and U.…”

Arctic boundaries of the ice model on three-bundle domains

“…This phenomenon was soon observed to be ubiquitous within the context of highly correlated statistical mechanical systems; see, for instance, [1,2,5,6,7,10,12,13,15,16,17,18,19,20,25,28,29,30,32,33,34,35,42,43,44,45,51,54,57,61]. In particular, Cohn-Kenyon-Propp developed a variational principle [12] that prescribes a law of large numbers for random domino tilings on almost arbitrary domains, which was used effectively by to explicitly determine the arctic boundaries of uniformly random lozenge tilings on polygonal domains.…”