Asymptotic domino statistics in the Aztec diamond

Chhita, Sunil; Johansson, Kurt; Young, Benjamin

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: Nov 3, 2016
• 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
• Public URL: https://durham-repository.worktribe.com/output/1372877