Skip to main content

Research Repository

Advanced Search

Domino statistics of the two-periodic Aztec diamond

Chhita, Sunil; Johansson, Kurt

Domino statistics of the two-periodic Aztec diamond Thumbnail


Kurt Johansson


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.


Chhita, S., & Johansson, K. (2016). Domino statistics of the two-periodic Aztec diamond. Advances in Mathematics, 294, 37-149.

Journal Article Type Article
Acceptance Date Feb 23, 2016
Online Publication Date Mar 9, 2016
Publication Date May 14, 2016
Deposit Date Oct 12, 2016
Publicly Available Date Mar 9, 2017
Journal Advances in Mathematics
Print ISSN 0001-8708
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 294
Pages 37-149


You might also like

Downloadable Citations