Local statistics for random domino tilings of the Aztec diamond
We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of...
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ftdatacite:10.48550/arxiv.math/0008243 2023-05-15T15:06:06+02:00 Local statistics for random domino tilings of the Aztec diamond Cohn, Henry Elkies, Noam Propp, James 2000 https://dx.doi.org/10.48550/arxiv.math/0008243 https://arxiv.org/abs/math/0008243 unknown arXiv https://dx.doi.org/10.1215/s0012-7094-96-08506-3 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 Probability math.PR FOS Mathematics Primary 60K35, 82B20, Secondary 05A16, 60C05 article-journal Article ScholarlyArticle Text 2000 ftdatacite https://doi.org/10.48550/arxiv.math/0008243 https://doi.org/10.1215/s0012-7094-96-08506-3 2022-04-01T16:56:36Z We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well. : 42 pages, 7 figures Text Arctic DataCite Metadata Store (German National Library of Science and Technology) Arctic Saddle Point ENVELOPE(73.483,73.483,-53.017,-53.017) |
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DataCite Metadata Store (German National Library of Science and Technology) |
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topic |
Combinatorics math.CO Probability math.PR FOS Mathematics Primary 60K35, 82B20, Secondary 05A16, 60C05 |
spellingShingle |
Combinatorics math.CO Probability math.PR FOS Mathematics Primary 60K35, 82B20, Secondary 05A16, 60C05 Cohn, Henry Elkies, Noam Propp, James Local statistics for random domino tilings of the Aztec diamond |
topic_facet |
Combinatorics math.CO Probability math.PR FOS Mathematics Primary 60K35, 82B20, Secondary 05A16, 60C05 |
description |
We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well. : 42 pages, 7 figures |
format |
Text |
author |
Cohn, Henry Elkies, Noam Propp, James |
author_facet |
Cohn, Henry Elkies, Noam Propp, James |
author_sort |
Cohn, Henry |
title |
Local statistics for random domino tilings of the Aztec diamond |
title_short |
Local statistics for random domino tilings of the Aztec diamond |
title_full |
Local statistics for random domino tilings of the Aztec diamond |
title_fullStr |
Local statistics for random domino tilings of the Aztec diamond |
title_full_unstemmed |
Local statistics for random domino tilings of the Aztec diamond |
title_sort |
local statistics for random domino tilings of the aztec diamond |
publisher |
arXiv |
publishDate |
2000 |
url |
https://dx.doi.org/10.48550/arxiv.math/0008243 https://arxiv.org/abs/math/0008243 |
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ENVELOPE(73.483,73.483,-53.017,-53.017) |
geographic |
Arctic Saddle Point |
geographic_facet |
Arctic Saddle Point |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
https://dx.doi.org/10.1215/s0012-7094-96-08506-3 |
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/0008243 https://doi.org/10.1215/s0012-7094-96-08506-3 |
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1766337751553146880 |