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The $q^{\mathrm{Volume}}$ lozenge tiling model via non-Hermitian orthogonal polynomials

Ahmad Barhoumi, Maurice Duits

Abstract

We consider the $q^\text{Volume}$ lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of non-intersecting paths. The starting point of our analysis is a formula for the correlation kernel due to Duits and Kuijlaars which involves the Christoffel-Darboux kernel of a particular family of non-Hermitian orthogonal polynomials. Our main results are split into two parts: the first part concerns the family of orthogonal polynomials, and the second concerns the behavior of the boundary of the so-called arctic curve. In the first half, we identify the orthogonal polynomials as a non-standard instance of little $q$-Jacobi polynomials and compute their large degree asymptotics in the $q \to 1$ regime. A consequence of this analysis is a proof that the zeros of the orthogonal polynomials accumulate on an arc of a circle and an asymptotic formula for the Christoffel-Darboux kernel. In the second half, we use these asymptotics to show that the boundary of the liquid region converges to the Airy process, in the sense of finite dimensional distributions, away from the boundary of the hexagon. At inflection points of the arctic curve, we show that we do not need to subtract/add a parabola to the Airy line ensemble, and this effect persists at distances which are $o(N^{-2/9})$ in the tangent direction.

The $q^{\mathrm{Volume}}$ lozenge tiling model via non-Hermitian orthogonal polynomials

Abstract

We consider the lozenge tiling model on a large, finite hexagon. It is well-known that random lozenge tilings of the hexagon correspond to a two-dimensional determinantal point process via a bijection with ensembles of non-intersecting paths. The starting point of our analysis is a formula for the correlation kernel due to Duits and Kuijlaars which involves the Christoffel-Darboux kernel of a particular family of non-Hermitian orthogonal polynomials. Our main results are split into two parts: the first part concerns the family of orthogonal polynomials, and the second concerns the behavior of the boundary of the so-called arctic curve. In the first half, we identify the orthogonal polynomials as a non-standard instance of little -Jacobi polynomials and compute their large degree asymptotics in the regime. A consequence of this analysis is a proof that the zeros of the orthogonal polynomials accumulate on an arc of a circle and an asymptotic formula for the Christoffel-Darboux kernel. In the second half, we use these asymptotics to show that the boundary of the liquid region converges to the Airy process, in the sense of finite dimensional distributions, away from the boundary of the hexagon. At inflection points of the arctic curve, we show that we do not need to subtract/add a parabola to the Airy line ensemble, and this effect persists at distances which are in the tangent direction.

Paper Structure

This paper contains 37 sections, 19 theorems, 288 equations, 19 figures.

Table of Contents

  1. Introduction
  2. General set-up
  3. Specialization to q-volume
  4. Statement of Results
  5. Identification of orthogonal polynomials
  6. Asymptotics of the orthogonal polynomials
  7. Asymptotics of the correlation kernel
  8. Overview of the rest of the paper
  9. Proof of Propositions \ref{['prop:jacobi']}
  10. Construction of the equilibrium measure and the g-function
  11. The density of the equilibrium measure
  12. Properties of the g-function
  13. Proof of Theorem \ref{['thm:polynomial-asymptotics']}
  14. Normalized Riemann-Hilbert problem
  15. Opening lenses
  16. ...and 22 more sections

Key Result

Proposition 2.1

Let $J_n(z; a, b|q)$ be as in eq:monic-jacobi and Then, $P_n(z)$ satisfies the orthogonality relation eq:ortho, where $\gamma$ is a contour going around the origin in the positive direction.

Figures (19)

  • Figure 1: Sample tiling of the hexagon with $N = 5$. All tiling images were generated using code kindly provided by Christophe Charlier.
  • Figure 2: Bijection between lozenge tilings and non-intersecting paths and the corresponding Directed, acyclic graph of a $4 \times 4 \times 4$ hexagon.
  • Figure 3: Sample tiling of the hexagon with $N = 5$ and the corresponding ensemble of non-intersecting paths.
  • Figure 4: Rotation of tiles corresponding to removing a box.
  • Figure 5: Sample tiling $\mathcal{T}$ and its complementary tiling $\widetilde{\mathcal{T}}$.
  • ...and 14 more figures

Theorems & Definitions (43)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 33 more