Random domino tilings and the arctic circle theorem
William Jockusch, James Gary Propp, Peter W. Shor · 2026 · 166 citationsRead the paper
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.
What the full text shows
Theory or simulation- The central sub-region of a random domino tiling of an Aztec diamond becomes arbitrarily close to a perfect circle as n increases. — radius n/√2 (n is sufficiently large; uniformly random tilings)
- Under a biased probability measure where horizontal tiles are chosen with probability p, the boundary of the temperate zone is an ellipse. — x²/p + y²/(1-p) = 1 (Gibbs state with energy -1/2 log p for horizontal dominoes)
- The number of domino tilings for an Aztec diamond of order n is exactly 2^(n(n+1)/2). — 2^(n(n+1)/2) tilings (Aztec diamond of order n)
- Setup
- Aztec diamonds of order n, including a specific example of order 64.
- How
- The authors use a stochastic 'domino shuffling' algorithm to generate uniform tilings and apply techniques from interacting particle systems, specifically the stationary behaviors of a totally asymmetric one-dimensional exclusion process in discrete time.
- Limitations
- The proof for the circular boundary is described as more complicated than desired. · Results are asymptotic (as n goes to infinity).
Summarized by Gemma 4 from the open-access full text · 9 Oct 2026. Read it