Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions· 2026Q2
Random domino tilings and the arctic circle theorem
- 160citations
- Q2SCImago
- 2026year
Short summary
For large Aztec diamonds, the central region of domino tilings closely approximates a circle of radius n/sqrt(2) in almost all cases.
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Key points
- Domino tilings of Aztec diamonds partition into five sub-regions.
- The central sub-region's shape approaches a circle of radius n/sqrt(2) for large n.
- This circular shape occurs in almost all tilings.
- The proof uses interacting particle systems and a specific exclusion process.
AI-generated from the title and abstract; the full text is not read.
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.
The authors' abstract, as published at the source. Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics