Random Domino Tilings and the Arctic Circle Theorem

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-...

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Bibliographic Details
Main Authors: Jockusch, William, Propp, James, Shor, Peter
Format: Report
Language:unknown
Published: arXiv 1998
Subjects:
Combinatorics math.CO
FOS Mathematics
Arctic
Online Access:https://dx.doi.org/10.48550/arxiv.math/9801068
https://arxiv.org/abs/math/9801068
id ftdatacite:10.48550/arxiv.math/9801068
record_format openpolar
spelling ftdatacite:10.48550/arxiv.math/9801068 2023-05-15T15:02:54+02:00 Random Domino Tilings and the Arctic Circle Theorem Jockusch, William Propp, James Shor, Peter 1998 https://dx.doi.org/10.48550/arxiv.math/9801068 https://arxiv.org/abs/math/9801068 unknown arXiv Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004 http://arxiv.org/licenses/assumed-1991-2003/ Combinatorics math.CO FOS Mathematics Preprint Article article CreativeWork 1998 ftdatacite https://doi.org/10.48550/arxiv.math/9801068 2022-04-01T16:53:17Z 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. : 37 pages of text plus 9 pages of figures (separate). [Note: This is not the final draft of this article.] Report Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic
institution Open Polar
collection DataCite Metadata Store (German National Library of Science and Technology)
op_collection_id ftdatacite
language unknown
topic Combinatorics math.CO
FOS Mathematics
spellingShingle Combinatorics math.CO
FOS Mathematics
Jockusch, William
Propp, James
Shor, Peter
Random Domino Tilings and the Arctic Circle Theorem
topic_facet Combinatorics math.CO
FOS Mathematics
description 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. : 37 pages of text plus 9 pages of figures (separate). [Note: This is not the final draft of this article.]
format Report
author Jockusch, William
Propp, James
Shor, Peter
author_facet Jockusch, William
Propp, James
Shor, Peter
author_sort Jockusch, William
title Random Domino Tilings and the Arctic Circle Theorem
title_short Random Domino Tilings and the Arctic Circle Theorem
title_full Random Domino Tilings and the Arctic Circle Theorem
title_fullStr Random Domino Tilings and the Arctic Circle Theorem
title_full_unstemmed Random Domino Tilings and the Arctic Circle Theorem
title_sort random domino tilings and the arctic circle theorem
publisher arXiv
publishDate 1998
url https://dx.doi.org/10.48550/arxiv.math/9801068
https://arxiv.org/abs/math/9801068
geographic Arctic
geographic_facet Arctic
genre Arctic
genre_facet Arctic
op_rights Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004
http://arxiv.org/licenses/assumed-1991-2003/
op_doi https://doi.org/10.48550/arxiv.math/9801068
_version_ 1766334815787810816

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