Asymptotic Geometry of Discrete Interlaced Patterns: Part II

Abstract

We study the asymptotic boundary of the liquid region of large random lozenge tiling models defined by uniformly random interlacing particle systems. In particular, we study a non-phase separating part of the boundary, i.e., a part of the boundary that does not border to a frozen phase. This is called the singular part of the boundary. We prove that isolated components of this boundary are lines and classify four different cases. Moreover, we show that the singular part of the boundary can have infinite one-dimensional Hausdorff measure. This has implications to the study of the free boundary problem arising in the variational problem studied by Kenyon and Okounkov and in a related work by De Silva and Savin.