Transversal Cusp-Airy versus Cusp-Airy for Lozenge Tilings

The fluctuations of lozenge tilings of hexagons with one or several cuts (nonconvexities) along opposite sides are governed by the (discrete-continuous) tacnode kernel ${\mathbb{L}}^{dTac}$, 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 $r\to \mathrm{\infty }$, 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 ($r\to \mathrm{\infty }$) of the tacnode kernel ${\mathbb{L}}^{dTac}$. 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.