Counting domino and lozenge tilings of reduced domains with Padé-type approximants

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 Padé 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-Padé approximants. For more complicated domains, such as hexagons with holes, the number of tilings involves a generalization of Hermite-Padé approximants.

💡 Research Summary

The paper introduces a novel analytic framework for evaluating gap probabilities in a broad class of discrete determinantal point processes whose kernels admit a double‑contour integral representation. This class encompasses the uniform measures on domino tilings of Aztec diamonds and lozenge tilings of hexagons, as well as their doubly‑periodic weighted extensions. The authors first simplify the kernels by means of a Fourier series expansion, which enables them to express the relevant partition functions as Fredholm determinants on ℓ²(ℤ).

A central contribution is the translation of these Fredholm determinants into Riemann–Hilbert (RH) problems. For the domino‑tiling case, the associated RH problem can be solved explicitly by a Padé‑type approximation: given the function
f_{m,j}^{N}(z;a)=z^{,m-j}(1-az)^{N-m+1},
the