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Domino statistics of the two-periodic Aztec diamond.

Chhita, Sunil and Johansson, Kurt (2016) 'Domino statistics of the two-periodic Aztec diamond.', Advances in mathematics., 294 . pp. 37-149.

Abstract

Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1016/j.aim.2016.02.025
Publisher statement:© 2016 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Date accepted:23 February 2016
Date deposited:23 November 2016
Date of first online publication:09 March 2016
Date first made open access:09 March 2017

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