Local statistics for random domino tilings of the Aztec diamond

Abstract

We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well.

Random Tilings of High Symmetry: II. Boundary Conditions and Numerical Studies

“…Our demonstration relies on the variational principle introduced in references [ 10,11 ] to characterize the entropy gradient between the center and the boundary in typical tilings with given fixed boundaries. Typical tilings are those which maximize an entropy functional defined as the integral over the tiling of a local entropy per tile; This local entropy is the free-boundary entropy calculated with the local fractions of tiles.…”

“…As already described in the introduction, such fixed boundary tilings lack a proper thermodynamic limit for finite D because the boundary has a strong macroscopic effect on the whole tiling 7,[9][10][11]14 . This leads to a spectacular effect known as the "arctic circle phenomenon 10 " in hexagonal (D = 3) tilings, where the tiling is periodic (and "frozen") outside a perfect circle at the large size limit and random inside this circle.…”

“…This leads to a spectacular effect known as the "arctic circle phenomenon 10 " in hexagonal (D = 3) tilings, where the tiling is periodic (and "frozen") outside a perfect circle at the large size limit and random inside this circle. More generally and for larger D, fixed boundary tilings have an effective codimension that is smaller near the boundary than in the bulk.…”

“…Although Computer Science is not our motivation, we are interested in the same enumeration problem 18,19 . More generally, the rich topological and combinatorial properties of random tilings make them an active field of research in pure mathematics, combinatorics and computer science 10,[20][21][22] . We shall use results from these fields throughout this paper.…”

Thermodynamic limit of the six-vertex model with domain wall boundary conditions

“…The "arctic circle theorem" [11] shows that as the size of the system grows large, the domino configurations become frozen outside the circle inscribed inside the diamond, and remain disordered but still heterogeneous [12] (i.e. non translationally invariant) inside the circle.…”