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Transversal Cusp-Airy versus Cusp-Airy for Lozenge Tilings

Mark Adler, Pierre van Moerbeke

TL;DR

This work analyzes fluctuations of lozenge tilings in nonconvex hexagons with two opposite cuts, where the large-size (or small-tiles) limit is governed by the discrete-tacnode kernel $\mathbb{L}^{\text{dTac}}$. Contrary to the natural expectation that cusp-Airy statistics would arise near the cusps, the authors uncover and characterize a new transversal cusp-Airy process ${\mathbb L}^{\text{T-cuspAiry}}$ that emerges in the $r\to\infty$ limit of ${\mathbb L}^{\text{dTac}}_{r,\rho,\beta}$ with $\rho=0$, $\beta=0$. The analysis blends Hermite polynomials, the GUE-kernel, integral operator identities, and delicate scaling arguments to decompose the tacnode kernel into ${\mathbb L}_0, {\mathbb L}_1, {\mathbb L}_2$ components and to track their asymptotics, establishing a precise edge-wise and cusp-wise universal limit. These results distinguish two distinct cusp-type universality classes (cusp-Airy vs transversal cusp-Airy) and connect them to geometric features (the $\{\rho\}$-strip vs the $\{\sigma\}$-strip), with implications for edge behavior and arctic curve cusps in nonconvex tilings.

Abstract

The fluctuations of lozenge tilings of hexagons with one or several cuts (nonconvexities) along opposite sides are governed by the (discrete-continuous) tacnode kernel ${\mathbb L}^{\mbox{\tiny dTac}}$, upon letting the hexagon become very large (or in other terms, keeping the hexagon fixed, with the tiles becoming very small). This is a point process with a finite number $r$ of (continuous) points along a discrete set of parallel lines within a specific region (see \cite{AJvM1,AJvM2}). Letting $r\to\infty$, one finds a liquid phase inscribed in the polygon, whose boundary (arctic curve) has a cusp near each cut, with two solid phases descending into the cusp (split-cusp). Duse-Johansson-Metcalfe \cite{DJM} show that in this situation the tile-fluctuations should obey the cusp-Airy statistics. It would have seem natural to expect to see the same cusp-Airy kernel in the neighborhood of the cut, for the limit ($r\to \infty$) of the tacnode kernel ${\mathbb L}^{\mbox{\tiny dTac}}$. As it turns out, another statistics appears: the {\em transversal cusp-Airy} statistics, which was a puzzling fact to all of us. This statistics is derived and fully explained in this paper.

Transversal Cusp-Airy versus Cusp-Airy for Lozenge Tilings

TL;DR

This work analyzes fluctuations of lozenge tilings in nonconvex hexagons with two opposite cuts, where the large-size (or small-tiles) limit is governed by the discrete-tacnode kernel . Contrary to the natural expectation that cusp-Airy statistics would arise near the cusps, the authors uncover and characterize a new transversal cusp-Airy process that emerges in the limit of with , . The analysis blends Hermite polynomials, the GUE-kernel, integral operator identities, and delicate scaling arguments to decompose the tacnode kernel into components and to track their asymptotics, establishing a precise edge-wise and cusp-wise universal limit. These results distinguish two distinct cusp-type universality classes (cusp-Airy vs transversal cusp-Airy) and connect them to geometric features (the -strip vs the -strip), with implications for edge behavior and arctic curve cusps in nonconvex tilings.

Abstract

The fluctuations of lozenge tilings of hexagons with one or several cuts (nonconvexities) along opposite sides are governed by the (discrete-continuous) tacnode kernel , upon letting the hexagon become very large (or in other terms, keeping the hexagon fixed, with the tiles becoming very small). This is a point process with a finite number of (continuous) points along a discrete set of parallel lines within a specific region (see \cite{AJvM1,AJvM2}). Letting , one finds a liquid phase inscribed in the polygon, whose boundary (arctic curve) has a cusp near each cut, with two solid phases descending into the cusp (split-cusp). Duse-Johansson-Metcalfe \cite{DJM} show that in this situation the tile-fluctuations should obey the cusp-Airy statistics. It would have seem natural to expect to see the same cusp-Airy kernel in the neighborhood of the cut, for the limit () of the tacnode kernel . As it turns out, another statistics appears: the {\em transversal cusp-Airy} statistics, which was a puzzling fact to all of us. This statistics is derived and fully explained in this paper.
Paper Structure (28 sections, 37 theorems, 220 equations)

This paper contains 28 sections, 37 theorems, 220 equations.

Table of Contents

  1. General introduction
  2. The tiling model, the discrete-tacnode kernel and the transversal cusp-Airy kernel
  3. Some background on tilings of nonconvex hexagons and the discrete tacnode kernel
  4. Hermite polynomials, the GUE-kernel and a useful identity
  5. Hermite polynomials
  6. GUE and its Fourier transform revisited
  7. A useful identity involving an integral operator ${\cal F}^{ \tau}_{ \xi}$ acting on $f(u,z)$
  8. Ratios of functions $\Theta^\pm_r$ in terms of the GUE-kernel for $\rho=0$
  9. The discrete-tacnode kernel for $\rho=0$
  10. The expression for the kernel ${\mathbb L}^{\hbox{\tiny dTac}}_{r,\rho,\beta}$ for $\rho=\beta=0$
  11. The ${\mathbb L}_2$-part of the kernel and Proof of Theorem \ref{['Th:prodop"']}
  12. Scaling and Asymptotics of the ${\mathbb L}_2$-part of the kernel
  13. Asymptotics of the ${\cal D}$-part
  14. Asymptotics of the ${\mathcal{E}}$-part
  15. Asymptotics of the ${\mathbb L}_{2 }$-term
  16. ...and 13 more sections

Key Result

Theorem 2.1

For $\rho=\beta=0$ and given ${\cal F}^{(\tau)}_\xi(u)$ as in (FR0) and $F^\xi_\tau(v)$ as in (tildeF), the discrete tacnode kernel ${\mathbb L}^{\hbox{\tiny dTac}}_{r,\rho,\beta}$ takes on the following form for arbitrary choices of $\varepsilon_i=\pm 1$: (using $\theta=\xi\sqrt2$ in the kernel (Fi

Theorems & Definitions (37)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Corollary 4.3
  • Lemma 5.1
  • Proposition 6.1
  • Lemma 6.2
  • ...and 27 more