Arctic circles, domino tilings and square Young tableaux
The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the “arctic cir-cle ” inscribed within the diamond. A similar arctic circle phenomenon has been observed in th...
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ftciteseerx:oai:CiteSeerX.psu:10.1.1.568.8297 2023-05-15T14:21:24+02:00 Arctic circles, domino tilings and square Young tableaux Dan Romik The Pennsylvania State University CiteSeerX Archives application/pdf http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.568.8297 en eng http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.568.8297 Metadata may be used without restrictions as long as the oai identifier remains attached to it. https://www.math.ucdavis.edu/~romik/data/uploads/papers/arctic-revised.pdf Key words and phrases Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform text ftciteseerx 2016-01-08T12:23:29Z The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the “arctic cir-cle ” inscribed within the diamond. A similar arctic circle phenomenon has been observed in the limiting behavior of random square Young tableaux. In this paper, we show that random domino tilings of the Aztec diamond are asymptotically related to random square Young tableaux in a more refined sense that looks also at the behavior in-side the arctic circle. This is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the Aztec diamond that uses the large-deviation techniques developed for the square Young tableaux problem in a previous paper by Pittel and the author. The solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square Young tableaux by Pittel and the author. The analytic techniques used to solve the variational problem provide a systematic, guess-free approach for solving problems of this type which have appeared in a number of related combinatorial probability models. Text Arctic Arctic Unknown Arctic |
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English |
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Key words and phrases Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform |
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Key words and phrases Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform Dan Romik Arctic circles, domino tilings and square Young tableaux |
topic_facet |
Key words and phrases Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform |
description |
The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the “arctic cir-cle ” inscribed within the diamond. A similar arctic circle phenomenon has been observed in the limiting behavior of random square Young tableaux. In this paper, we show that random domino tilings of the Aztec diamond are asymptotically related to random square Young tableaux in a more refined sense that looks also at the behavior in-side the arctic circle. This is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the Aztec diamond that uses the large-deviation techniques developed for the square Young tableaux problem in a previous paper by Pittel and the author. The solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square Young tableaux by Pittel and the author. The analytic techniques used to solve the variational problem provide a systematic, guess-free approach for solving problems of this type which have appeared in a number of related combinatorial probability models. |
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The Pennsylvania State University CiteSeerX Archives |
format |
Text |
author |
Dan Romik |
author_facet |
Dan Romik |
author_sort |
Dan Romik |
title |
Arctic circles, domino tilings and square Young tableaux |
title_short |
Arctic circles, domino tilings and square Young tableaux |
title_full |
Arctic circles, domino tilings and square Young tableaux |
title_fullStr |
Arctic circles, domino tilings and square Young tableaux |
title_full_unstemmed |
Arctic circles, domino tilings and square Young tableaux |
title_sort |
arctic circles, domino tilings and square young tableaux |
url |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.568.8297 |
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Arctic |
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Arctic |
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Arctic Arctic |
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Arctic Arctic |
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https://www.math.ucdavis.edu/~romik/data/uploads/papers/arctic-revised.pdf |
op_relation |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.568.8297 |
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Metadata may be used without restrictions as long as the oai identifier remains attached to it. |
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