Dimer models on astroidal zig-zag graphs

TL;DR

This paper constructs explicit inverse Kasteleyn matrices for a new family of periodic bipartite graphs called astroidal zig-zag graphs, enabling detailed analysis of their dimer models and phase behavior.

Contribution

It introduces a novel class of graphs with explicit inverse Kasteleyn matrices and analyzes their phase separation and limit shape in the dimer model.

Findings

Explicit inverse Kasteleyn matrices for AZ graphs are given by double contour integrals.

AZ graphs exhibit phase separation into frozen, rough, and smooth regions.

The arctic curve and limit shape are explicitly parametrized and proven to converge.

Abstract

On a finite weighted graph, the dimer model is a probability measure on its dimer covers, that assigns to any cover a probability proportional to the product of the weights of its edges. For planar bipartite graphs, dimer correlations are encoded by the inverse of the so-called Kasteleyn matrix; for a large graph, typically taken as a finite domain in a periodic graph, this inverse matrix is known explicitly only for a handful of examples. In all previously known examples, the Newton polygon -- a convex lattice polygon that classifies periodic graphs up to local moves -- is either a triangle or a quadrilateral. Our main results are the following. For any (minimal) periodic planar bipartite graph, we construct an $(n-3)$-dimensional family of finite subgraphs for which we obtain an explicit inverse Kasteleyn matrix; here $n$ is the number of sides of the Newton polygon. Their boundaries are formed by zig-zag paths and their overall shape is reminiscent of an astroid; we call them astroidal zig-zag graphs (AZ graphs). If the Newton polygon is the unit square then the corresponding AZ graph is the celebrated Aztec diamond with its size as the parameter. Our inverse Kasteleyn matrices are given by a double contour integral on the corresponding spectral curve for any Fock weighting of the graph. This includes, in particular, all periodic weightings. For periodic weightings, we asymptotically analyze the resulting inverse Kasteleyn matrices. We establish a phase separation in large AZ graphs into asymptotically frozen, rough (liquid), and smooth (gaseous) regions, and obtain an explicit parametrization of the `arctic curve'. We also compute the deterministic limit of the height function, known as the limit shape, and prove the convergence of the local dimer correlations to the translation-invariant Gibbs measure of the slope predicted by the limit shape.