Counting domino and lozenge tilings of reduced domains with Pad\'{e}-type approximants

TL;DR

This paper develops a novel method using Riemann-Hilbert problems and Padé-type approximants to explicitly count domino and lozenge tilings of reduced domains, including Aztec diamonds and hexagons.

Contribution

It introduces a new approach combining Fourier series and Riemann-Hilbert problems to derive explicit tiling counts via Padé approximants.

Findings

Explicit formulas for domino tilings of reduced Aztec diamonds.

Explicit formulas for lozenge tilings of reduced hexagons.

Extension to domains with holes involving generalized Hermite-Padé approximants.

Abstract

We introduce a new method for studying gap probabilities in a class of discrete determinantal point processes with double contour integral kernels. This class of point processes includes uniform measures of domino and lozenge tilings as well as their doubly periodic generalizations. We use a Fourier series approach to simplify the form of the kernels and to characterize gap probabilities in terms of Riemann-Hilbert problems. As a first illustration of our approach, we obtain an explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Pad\'e approximants, by solving the associated Riemann-Hilbert problem explicitly. As a second application, we obtain an explicit expression for the number of lozenge tilings of (simply connected) reduced hexagons in terms of Hermite-Pad\'e approximants. For more complicated domains, such as hexagons with holes, the number of tilings involves a generalization of Hermite-Pad\'e approximants.