A variational principle for domino tilings of multiply-connected domains

TL;DR

This paper establishes a variational principle for domino tilings in multiply-connected domains, proving large deviation principles and a law of large numbers for the height function in different asymptotic regimes.

Contribution

It introduces a variational framework for domino tilings in complex domains and proves large deviation principles and a law of large numbers for the height function.

Findings

Large deviation principle for the height function in two regimes

Law of large numbers for height change in the first regime

Characterization of domino tilings with specified asymptotic height change

Abstract

We study random domino tilings of a multiply-connected domain with a height function defined on the universal covering space of the domain. We prove a large deviation principle for the height function in two asymptotic regimes. The first regime covers all domino tilings of the domain. We also prove a law of large numbers for height change in this regime. The second regime covers domino tilings with a given asymptotic height change $r$.