Random Domino Tilings and the Arctic Circle Theorem

Abstract

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.

Figures (9)

• Figure 1: A random domino tiling of the Aztec diamond of order 64.
• Figure 2: Domino shuffling, before dominoes slide.
• Figure 3: Domino shuffling, after dominoes slide.
• Figure 4: The arctic region of a tiling.
• Figure 5: Three cases.