Abstract : An $N$ dimer cover of a graph is a collection of edges such that every vertex is contained in exactly $N$ edges of the collection. The multinomial dimer model studies a family of natural but non-uniform measures on $N$ dimer covers. In the large $N$ limit, this model turns out to be exactly solvable in a strong sense, in any dimension $N$. In this talk, I will define the model, and discuss its properties on subgraphs of lattices in the iterated limit as the multiplicity $N$ and then the size of the graph go to infinity, analogous to the scaling limit question for 2D standard dimers addressed by Cohn, Kenyon, and Propp. In this setting we can explicitly compute limit shapes in some examples, in particular for the Aztec diamond and a 3D analog called the Aztec cuboid. I will also discuss the surrounding theory, including explicit formulas for the free energy, large deviations, EulerLagrange equations, gauge functions, and regularity properties of limit shapes.This is joint work with Rick Kenyon.

Keywords : dimer model; large N limit; domino tilings; variational principle; large deviations; limit shape; critical gauge