Transversal Cusp-Airy versus Cusp-Airy for Lozenge Tilings

TL;DR

This paper investigates the complex boundary fluctuations in lozenge tilings of hexagons with cuts, revealing a new transversal cusp-Airy statistics that differs from the expected cusp-Airy behavior near the cusps.

Contribution

The paper introduces and explains the novel transversal cusp-Airy statistics observed in lozenge tilings, expanding understanding of boundary fluctuation behaviors.

Findings

Identification of transversal cusp-Airy statistics in lozenge tilings

Distinction between cusp-Airy and transversal cusp-Airy behaviors

Explanation of the emergence of new fluctuation statistics

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.