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 circle” inscribed within the diamond. A similar arctic circle phenomenon has been observed in the...
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ftculeuclid:oai:CULeuclid:euclid.aop/1332772715 2023-05-15T14:36:25+02:00 Arctic circles, domino tilings and square Young tableaux Romik, Dan 2012-03 application/pdf https://projecteuclid.org/euclid.aop/1332772715 https://doi.org/10.1214/10-AOP628 en eng The Institute of Mathematical Statistics 0091-1798 2168-894X https://projecteuclid.org/euclid.aop/1332772715 Ann. Probab. 40, no. 2 (2012), 611-647 doi:10.1214/10-AOP628 Copyright 2012 Institute of Mathematical Statistics Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform 60C05 60K35 60F10 Text 2012 ftculeuclid https://doi.org/10.1214/10-AOP628 2019-12-22T01:08:50Z 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 circle” 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 inside 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 Project Euclid (Cornell University Library) Arctic The Annals of Probability 40 2 |
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Project Euclid (Cornell University Library) |
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ftculeuclid |
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English |
topic |
Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform 60C05 60K35 60F10 |
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Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform 60C05 60K35 60F10 Romik, Dan Arctic circles, domino tilings and square Young tableaux |
topic_facet |
Domino tiling Young tableau alternating sign matrix Aztec diamond arctic circle large deviations variational problem combinatorial probability Hilbert transform 60C05 60K35 60F10 |
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 circle” 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 inside 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. |
format |
Text |
author |
Romik, Dan |
author_facet |
Romik, Dan |
author_sort |
Romik, Dan |
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 |
publisher |
The Institute of Mathematical Statistics |
publishDate |
2012 |
url |
https://projecteuclid.org/euclid.aop/1332772715 https://doi.org/10.1214/10-AOP628 |
geographic |
Arctic |
geographic_facet |
Arctic |
genre |
Arctic |
genre_facet |
Arctic |
op_relation |
0091-1798 2168-894X https://projecteuclid.org/euclid.aop/1332772715 Ann. Probab. 40, no. 2 (2012), 611-647 doi:10.1214/10-AOP628 |
op_rights |
Copyright 2012 Institute of Mathematical Statistics |
op_doi |
https://doi.org/10.1214/10-AOP628 |
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The Annals of Probability |
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40 |
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2 |
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1766309037641564160 |