Random Domino Tilings and the Arctic Circle Theorem

1998/01/13 by William Jockusch, James Propp, Jockusch, William +3 · 1 voice · 11 citations
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.math/9801068

arxiv published 1998/01/13 · arxiv updated 2026/04/07

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.

Cited by

• Quenched and Annealed CLTs for the one-periodic Aztec diamond in random environment
• Limit shapes for Domain-Wall (colored) vertex models
• GUE-corners process in two-periodic Aztec diamonds
• Frozen-corner enumeration of Alternating Sign Matrices
• Emergent Hydrodynamics in an Exclusion Process with Long-Range Interactions
• Domino tilings beyond 2D
• Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions
• Domain-wall melting and entanglement in free-fermion chains with a band structure
• Breakdown of the thermodynamic limit in quantum spin and dimer models
• Two dimensional inhomogeneous classical systems at criticality
• The q^Volume lozenge tiling model via non-Hermitian orthogonal polynomials

Discussions

• ohh clearer explanation of the aztec diamond dimer cover shuffling algorithm here [bsky, 0 points, 0 comments]

Related

• Local statistics for random domino tilings of the Aztec diamond
• Asymptotics of random domino tilings of rectangular Aztec diamonds
• Boundary dents, the arctic circle and the arctic ellipse
• An arctic circle theorem for groves
• Fluctuations and arctic curve in the Aztec diamond