Random Domino Tilings and the Arctic Circle Theorem

In this article we study domino tilings of a family of finite regions called Aztec diamonds. Every such tiling determines a partition of the Aztec diamond into five sub-regions; in the four outer sub-regions, every tile lines up with nearby tiles, while in the fifth, central sub-region, differently-oriented tiles co-exist side by side. We show that when n is sufficiently large, the shape of the central sub-region becomes arbitrarily close to a perfect circle of radius n/sqrt(2) for all but a negligible proportion of the tilings. Our proof uses techniques from the theory of interacting particle systems. In particular, we prove and make use of a classification of the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.

💡 Research Summary

The paper provides a self‑contained probabilistic proof of the Arctic Circle Theorem for domino tilings of Aztec diamonds. An Aztec diamond of order n is a planar region consisting of 2 n (n + 1) unit squares arranged in a stepped shape; a domino is a 2 × 1 or 1 × 2 rectangle covering two adjacent squares. Every domino tiling of such a region can be equipped with a “heading” (north, south, east, west) by marking a distinguished vertex on each domino; this heading will later be used to define a combinatorial operation called shuffling.

The shuffling algorithm transforms a tiling of order n − 1 into a tiling of order n in three steps: (1) destruction, where all “bad blocks” (adjacent dominoes whose arrows point toward each other) are removed; (2) sliding, where each remaining domino moves one unit in the direction of its arrow; (3) creation, where the resulting 2 × 2 holes are filled with either a horizontal or a vertical domino pair, chosen uniformly at random. Repeating shuffling n times starting from the empty tiling of order 0 yields a uniformly random tiling of order n; each of the 2^{n(n+1)/2} tilings appears with probability 2^{-n(n+1)/2}.

The main result, Theorem 1 (the Arctic Circle Theorem), states that for any ε > 0, when n is sufficiently large, at least a (1 − ε) fraction of all tilings have a “temperate zone” whose boundary stays within distance ε n of the inscribed circle of radius n/√2 centred in the diamond. Equivalently, the symmetric difference between the temperate zone and the interior of that circle has area o(n²). This phenomenon is striking because the same region tiled by a plain 2n × 2n square exhibits no macroscopic frozen pattern.

The proof reduces the geometric statement to a probabilistic statement about a one‑dimensional interacting particle system: the totally asymmetric simple exclusion process (TASEP) in discrete time. In this process, a configuration is an infinite binary sequence …,x_{−1},x_0,x_1,… where x_i=1 indicates a particle at site i. At each time step every particle that has an empty neighbour to its right independently attempts to jump right with probability ½; if the neighbour is occupied the particle cannot move. Starting from the step initial condition x*_i = 1 for i ≤ 0 and 0 otherwise, Theorem 2 asserts that the number of particles in the interval