Asymptotic Geometry of Discrete Interlaced Patterns: Part II

Abstract

We study the boundary of the liquid region $\mathcal{L}$ in large random lozenge tiling models defined by uniform random interlacing particle systems with general initial configuration, which lies on the line $(x,1)$, $x\in\mathbb{R}\equiv \partial \mathbb{H}$. We assume that the initial particle configuration converges weakly to a limiting density $\phi(x)$, $0\le \phi\leq 1$. The liquid region is given by a homeomorphism $W_{\mathcal{L}}: \mathcal{L}\to \mathbb{H}$, the upper half plane, and we consider the extension of $W_{\mathcal{L}}^{-1}$ to $\overline{\mathbb{H}}$. Part of $\partial \mathcal{L}$ is given by a curve, the edge $\mathcal{E}$, parametrized by intervals in $\partial \mathbb{H}$, and this corresponds to points where $\phi$ is identical to $0$ or $1$. If $0<\phi<1$, the non-trivial support, there are two cases. Either $W_{\mathcal{L}}^{-1}(w)$ has the limit $(x,1)$ as $w\to x$ non-tangentially and we have a \emph{regular point}, or we have what we call a singular point. In this case $W_{\mathcal{L}}^{-1}$ does not extend continuously to $\overline{\mathbb{H}}$. Singular points give rise to parts of $\partial \mathcal{L}$ not given by $\mathcal{E}$ and which can border a frozen region, or be "inside" the liquid region. This shows that in general the boundary of $\partial \mathcal{L}$ can be very complicated. We expect that on the singular parts of $\partial \mathcal{L}$ we do not get a universal point process like the Airy or the extended Sine kernel point processes. Furthermore, $\mathcal{E}$ and the singular parts of $\partial \mathcal{L}$ are shocks of the complex Burgers equation.