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

Christophe Charlier, Tom Claeys

TL;DR

This work develops a unified Fourier–Riemann–Hilbert framework to study gap probabilities in discrete determinantal point processes associated with domino tilings of reduced Aztec diamonds and lozenge tilings of reduced hexagons. By representing gap events as Fredholm determinants, conjugating to integrable kernels, and solving a corresponding RH problem, the authors connect gap probabilities to Padé and Hermite–Padé approximants. The main results yield explicit expressions for tiling counts in reduced domains: F_N^{m,k}(a) in terms of Padé data κ_N^{m,j}(a) and G_{L,M,N}^{r,k} in terms of Hermite–Padé data, with one-gap and semi-infinite cases recovering concrete formulas via RH solutions. This methodology not only provides exact finite-N formulas but also establishes a pathway for asymptotic analysis of tilings with holes or reductions, bridging combinatorics, integrable systems, and approximation theory. The framework is poised to extend to more general multi-periodic tilings and to yield detailed large-gap asymptotics in tiling models.

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 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.

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

TL;DR

This work develops a unified Fourier–Riemann–Hilbert framework to study gap probabilities in discrete determinantal point processes associated with domino tilings of reduced Aztec diamonds and lozenge tilings of reduced hexagons. By representing gap events as Fredholm determinants, conjugating to integrable kernels, and solving a corresponding RH problem, the authors connect gap probabilities to Padé and Hermite–Padé approximants. The main results yield explicit expressions for tiling counts in reduced domains: F_N^{m,k}(a) in terms of Padé data κ_N^{m,j}(a) and G_{L,M,N}^{r,k} in terms of Hermite–Padé data, with one-gap and semi-infinite cases recovering concrete formulas via RH solutions. This methodology not only provides exact finite-N formulas but also establishes a pathway for asymptotic analysis of tilings with holes or reductions, bridging combinatorics, integrable systems, and approximation theory. The framework is poised to extend to more general multi-periodic tilings and to yield detailed large-gap asymptotics in tiling models.

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 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.

Paper Structure

This paper contains 24 sections, 13 theorems, 149 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Domino tilings of reduced Aztec diamonds
  3. Domino tilings of the Aztec diamond.
  4. Domino tilings of reduced Aztec diamonds and Padé approximants.
  5. Lozenge tilings of reduced hexagons
  6. Lozenge tilings of hexagons.
  7. Lozenge tilings of reduced hexagons and Hermite-Padé approximation.
  8. General methodology
  9. Step 1: Fredholm determinant representation.
  10. Step 2: Fourier series conjugation.
  11. Step 3: RH characterization.
  12. Step 4: Solve the RH problem explicitly in terms of a Padé approximation problem.
  13. Tilings and non-intersecting paths
  14. Domino tilings of the Aztec diamond
  15. Lozenge tilings of hexagons
  16. ...and 9 more sections

Key Result

Theorem 1.1

For $N\in \mathbb{N}_{>0}:=\{1,2,\ldots\}$, $m\in\{1,\ldots, N \}$, $k\in\{1,\ldots, m+1\}$, $0<a\leq 1$, we have In particular, the number of tilings of $A_N^{m,k}$ is equal to $2^{\frac{N(N+1)}{2}}\prod_{j=k}^{m}\frac{1}{\kappa_N^{m,j}(1)}$.

Figures (17)

  • Figure 1: Left: a tiling of $A_{5}$. The north, south, east and west dominoes are shown in red, yellow, green and blue, respectively. Right: a tiling of $A_{300}$ chosen uniformly at random.
  • Figure 2: Left: a tiling of $A_{7}^{5,2}$. The north, south, east and west dominoes are shown in red, yellow, green and blue, respectively. The set $A_{N}\setminus A_{N}^{m,k}$ is shown in grey; it contains $N+1-m$ corners on the north-west side and $m-k+1$ corners on the north-east side. Right: a tiling of $A_{300}^{150,80}$ chosen uniformly at random.
  • Figure 3: Any tiling of $A_{N}^{m,1}$ decouples into a tiling of an Aztec diamond of order $m-1$ and an Aztec diamond of order $N-m$, whose boundaries are shown with thick black lines. The set $A_{N}\setminus A_{N}^{m,k}$ is shown in grey, and its mirror image with respect to $y=0$ is tiled only with south dominoes. Left: a tiling of $A_{10}^{7,1}$. Right: a tiling of $A_{300}^{70,1}$ chosen uniformly at random.
  • Figure 4: Left: a tiling of $A_{200}^{50,2}$ chosen uniformly at random. Right: a tiling of $A_{200}^{50,3}$ chosen uniformly at random.
  • Figure 5: Left: a tiling of $H_{L,M,N}$ with $L=11$, $M=5$ and $N=4$. Right: a tiling of $H_{200,100,100}$ chosen uniformly at random.
  • ...and 12 more figures

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 16 more