Asymptotic Geometry of Discrete Interlaced Patterns: Part II

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 $(x,1)$, $x\in \mathbb{R}\equiv \mathrm{\partial }\mathbb{H}$. We assume that the initial particle configuration converges weakly to a limiting density $\varphi (x)$, $0\le \phi\leq 1$\mathrm{.}Theliquidregionisgivenbyahomeomorphism$W_{$\mathcal{L}$}: $\mathcal{L}$\to $\mathbb{H}$$, the upper half plane, and we consider the extension of $W_{\mathcal{L}}^{-1}$ to $\bar{\mathbb{H}}$. Part of $\partial $\mathcal{L}$$isgivenbyacurve,theedge$$\mathcal{E}$$, parametrized by intervals in $\mathrm{\partial }\mathbb{H}$, and this corresponds to points where $\varphi$ is identical to $0$ or $1$. If $0<\varphi <1$, 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 $\bar{\mathbb{H}}$. Singular points give rise to parts of $\mathrm{\partial }\mathcal{L}$ 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 $\mathrm{\partial }\mathcal{L}$ can be very complicated. We expect that on the singular parts of $\mathrm{\partial }\mathcal{L}$ 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 $\mathrm{\partial }\mathcal{L}$ are shocks of the complex Burgers equation.