Dimers and amoebae

Abstract

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.

Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

“…In fact any model with short range interactions and a nonzero facet edge curvature is expected to be in the universality class discussed here. There are other surface models which are still determinantal and exhibit facets in equilibrium [33,31]. To establish that their fluctuation properties are determined by GUE random matrix theory remains as a task for the future.…”

Dimers and orientifolds

“…Given a dimer diagram, one can define zigzag paths (these, along with the related JHEP09(2007)075 rhombi paths, were introduced in the mathematical literature on dimers in [31,32], and applied to the quiver gauge theory context in [21]), as paths composed of edges, and which turn maximally to the right at e.g. black nodes and maximally to the left at white nodes.…”

Crystal melting and black holes

“…By suitably rescaling the ranks and locations of quiver nodes, we obtain a limit shape for the profile of the molten crystal [45,46,10] which defines the charges of an average BPS black hole. In the immediate vicinity of a quiver node all of its neighboring ranks are equal.…”

“…This formal difference determines a homology 1-cycle in the T 2 of the dimer model. As shown in [10], the corresponding subset of lattice points in H 1 (T 2 , Z) ≃ Z×Z defines a convex polytope. We define an internal (external) perfect matching as one where its lattice point lies in the interior (boundary) of this polytope.…”

“…To describe heights for 3d crystals, we use dimer models. Recent reviews on the mathematical physics of dimer models are given in [9,10]. A dimer model corresponds to a collection of black and white atoms on a bipartite lattice 1 .…”