The Annals of Applied Probability

Asymptotic domino statistics in the Aztec diamond

Sunil Chhita, Kurt Johansson, and Benjamin Young

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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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 3 (2015), 1232-1278.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124128

Digital Object Identifier
doi:10.1214/14-AAP1021

Mathematical Reviews number (MathSciNet)
MR3325273

Zentralblatt MATH identifier
1329.60331

Subjects
Primary: 60G55: Point processes
Secondary: 60C05: Combinatorial probability

Keywords
Aztec diamond domino tiling dimer covering determinantal point process

Citation

Chhita, Sunil; Johansson, Kurt; Young, Benjamin. Asymptotic domino statistics in the Aztec diamond. Ann. Appl. Probab. 25 (2015), no. 3, 1232--1278. doi:10.1214/14-AAP1021. https://projecteuclid.org/euclid.aoap/1427124128


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