Asymptotic domino statistics in the Aztec diamond

TL;DR

This paper analyzes the asymptotic behavior of domino tilings in the Aztec diamond, revealing new connections between correlation kernels and inverse Kasteleyn matrices, and describing the limiting processes near boundaries.

Contribution

It provides a generalized formula for the inverse Kasteleyn matrix and characterizes the asymptotic domino processes near the frozen boundary with new point process descriptions.

Findings

Southern dominoes near the frozen boundary converge to a thinned Airy point process.

Holes of the southern domino process converge to a thickened Airy point process.

Domino process in the unfrozen region converges to the limiting Gibbs measure.

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.