Domino tilings of the Aztec diamond with doubly periodic weightings

TL;DR

This paper analyzes domino tilings of the Aztec diamond with doubly periodic weights, deriving local and macroscopic behaviors, including arctic curves and height functions, using integral formulas and steepest descent methods.

Contribution

It introduces a double integral formula for the correlation kernel and studies the asymptotic behavior for models with multiple smooth regions, confirming the complex Burgers' equation in the rough region.

Findings

Derived the correlation kernel for finite Aztec diamond

Identified arctic curves and smooth regions in the limit

Confirmed the complex Burgers' equation for the height function

Abstract

In this paper we consider domino tilings of the Aztec diamond with doubly periodic weightings. In particular a family of models which, for any $ k \in \mathbb{N} $, includes models with $ k $ smooth regions is analyzed as the size of the Aztec diamond tends to infinity. We use a non-intersecting paths formulation and give a double integral formula for the correlation kernel of the Aztec diamond of finite size. By a classical steepest descent analysis of the correlation kernel we obtain the local behavior in the smooth and rough regions as the size of the Aztec diamond tends to infinity. From the mentioned limit the macroscopic picture such as the arctic curves and in particular the number of smooth regions is deduced. Moreover we compute the limit of the height function and as a consequence we confirm, in the setting of this paper, that the limit in the rough region fulfills the complex Burgers' equation, as stated by Kenyon and Okounkov.