Open Access
March 2012 Arctic circles, domino tilings and square Young tableaux
Dan Romik
Ann. Probab. 40(2): 611-647 (March 2012). DOI: 10.1214/10-AOP628


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.


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Dan Romik. "Arctic circles, domino tilings and square Young tableaux." Ann. Probab. 40 (2) 611 - 647, March 2012.


Published: March 2012
First available in Project Euclid: 26 March 2012

zbMATH: 1258.60014
MathSciNet: MR2952086
Digital Object Identifier: 10.1214/10-AOP628

Primary: 60C05 , 60F10 , 60K35

Keywords: alternating sign matrix , arctic circle , Aztec diamond , combinatorial probability , domino tiling , Hilbert transform , large deviations , variational problem , Young tableau

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 2 • March 2012
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