Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 26 min ago
Peer-reviewedCC BYAnnales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions

Random domino tilings and the arctic circle theorem

  • William Jockusch1,2
  • James Gary Propp3,4
  • Peter W. Shor5,6
6 affiliations
  1. 1SC-B Consulting, Urbana, USA
  2. 2Wizards of the Coast
  3. 3University of Massachusetts Lowell, USA
  4. 4MIT
  5. 5Massachusetts Institute of Technology, Cambridge, USA
  6. 6AT&T-Labs
Open at sourceDOICite

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.

Related

  1. Explicit finite size corrections for Toeplitz determinants and rectangular domino tilings
    Miguel Tierz
    arXivPreprintarXiv licenceFull textAnalysis & PDEsCombinatorics & discrete mathematicsarXiv
  2. Billiard Orbits in Young Diagrams: Medial Links, Bicycle Spaces, and Domino Tilings
    David J. Hemmer
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. Parity-dependent sublattice structure of local 2×2 correlations in domino tilings of the Aztec diamond
    Chaeyeon Kim, Suhwan Yoo, Jeehoon Choi +2
    arXivPreprintarXiv licenceFull textMathematical physicsStatistical & nonlinear physicsarXiv
  4. Polyhedral normed spaces: the structural theorem and locally finite tilings
    Carlo Alberto De Bernardi, Helena del Rìo, Tommaso Russo +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Random Fibonacci Tilings
    Yuval Amit, Azmera Gebre, Christopher Housholder +4
    arXivPreprintarXiv licenceFull textProbabilityarXiv

Cite

BibTeXRIS
BibTeX
@article{jockusch2026random,
  title = {{Random domino tilings and the arctic circle theorem}},
  author = {William Jockusch and James Gary Propp and Peter W. Shor},
  journal = {Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions},
  year = {2026},
  month = may,
  doi = {10.4171/aihpd/233},
  url = {https://doi.org/10.4171/aihpd/233},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Random domino tilings and the arctic circle theorem
AU  - William Jockusch
AU  - James Gary Propp
AU  - Peter W. Shor
PY  - 2026
DA  - 2026/05/11
JO  - Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions
DO  - 10.4171/aihpd/233
UR  - https://doi.org/10.4171/aihpd/233
AB  - 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.
SN  - 2308-5827
SN  - 2308-5835
ER  -