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Asymptotic domino statistics in the Aztec diamond

Chhita, Sunil; Johansson, Kurt; Young, Benjamin

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Authors

Kurt Johansson

Benjamin Young



Abstract

We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

Citation

Chhita, S., Johansson, K., & Young, B. (2015). Asymptotic domino statistics in the Aztec diamond. Annals of Applied Probability, 25(3), 1232-1278. https://doi.org/10.1214/14-aap1021

Journal Article Type Article
Online Publication Date Mar 23, 2015
Publication Date Jun 1, 2015
Deposit Date Oct 12, 2016
Publicly Available Date Mar 28, 2024
Journal Annals of Applied Probability
Print ISSN 1050-5164
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 25
Issue 3
Pages 1232-1278
DOI https://doi.org/10.1214/14-aap1021

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