Asymptotics of random domino tilings of rectangular Aztec diamonds

TL;DR

This paper studies the asymptotic behavior of domino tilings on rectangular Aztec diamonds, proving a law of large numbers, describing the frozen boundary, and showing fluctuations converge to the Gaussian Free Field.

Contribution

It introduces explicit formulas for the limit shape and frozen boundary of domino tilings on rectangular Aztec diamonds, advancing understanding of their asymptotic properties.

Findings

Proved Law of Large Numbers for height functions.

Derived explicit formulas for the limit shape and frozen boundary.

Established convergence of fluctuations to the Gaussian Free Field.

Abstract

We consider asymtotics of a domino tiling model on a class of domains which we call rectangular Aztec diamonds. We prove the Law of Large Numbers for the corresponding height functions and provide explicit formulas for the limit. For a special class of examples, the explicit parametrization of the frozen boundary is given. It turns out to be an algebraic curve with very special properties. Moreover, we establish the convergence of the fluctuations to the Gaussian Free Field in appropriate coordinates. Our main tool is a recently developed moment method for discrete particle systems.