Dimer models and conformal structures

Abstract

Dimer models have been the focus of intense research efforts over the last years. Our paper grew out of an effort to develop new methods to study minimizers or the asymptotic height functions of general dimer models and the geometry of their frozen boundaries. We prove a complete classification of the regularity of minimizers and frozen boundaries for all dimer models for a natural class of polygonal domains, much studied in numerical simulations and elsewhere. In particular, we show that the frozen boundaries are always algebraic curves. Our classification also implies that the Pokrovsky‐Talapov law holds for all dimer models at a generic point on the frozen boundary and, in addition, shows a very strong local rigidity of dimer models, which can be interpreted as a geometric universality result. Indeed, we prove a converse result, showing that any geometric situation for any dimer model is, in the simply connected case, realized already by the lozenge model. To achieve these goals we develop a new study on the boundary regularity for a class of Monge–Ampère equations in non‐strictly convex domains, of independent interest, as well as a new approach to minimality for a general dimer functional. In the context of polygonal domains, we give the first general results for the existence of gas domains for minimizers.