Domino statistics of the two-periodic Aztec diamond

TL;DR

This paper simplifies the analysis of the two-periodic Aztec diamond domino tilings by deriving a double contour integral formula for the inverse Kasteleyn matrix, enabling asymptotic analysis across phases and boundaries.

Contribution

It introduces a simplified double contour integral formula for the inverse Kasteleyn matrix in the two-periodic Aztec diamond model, facilitating detailed asymptotic analysis.

Findings

Convergence of inverse Kasteleyn matrix entries to full-plane limits in different phases.

Identification of the extended Airy kernel at phase boundaries.

Potential combinatorial description of the liquid-gas boundary.

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.