Random Domino Tilings and the Arctic Circle Theorem

William Jockusch; James Propp; Peter Shor

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arXiv · math/9801068

Random Domino Tilings and the Arctic Circle Theorem

William Jockusch·James Propp·Peter Shor

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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.

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William Jockusch, James Propp, Peter Shor. 2026-04-07. Random Domino Tilings and the Arctic Circle Theorem. https://arxiv.org/abs/math/9801068

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