The multinomial dimer model

The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large $N$ limits in lattice gauge theory, we study a large $N$ limit of the dimer model in any dimension $d$. The dependence on $N$ comes from the multinomial tiling model introduced by Kenyon and Pohoata, which gives a general framework for adding a dependence on $N$ to a tiling model. We study the behavior of this model on periodic bipartite graphs in ${\mathbb R}^d$, in the scaling limit as the multiplicity $N$ and then the size of the graph go to infinity. In this iterated limit, in any dimension $d$, we prove a variational principle and show that random configurations concentrate on a limit shape which is the unique solution to an associated system of Euler-Lagrange equations. The rate function of the variational principle is the integral of a surface tension function, which we can compute explicitly for lattices in any dimension $d$ as the Legendre dual of the free energy for the model on the torus. We give a unified methodology for computing the surface tension and Euler-Lagrange equations in any dimension $d$. A new structure called the critical gauge also emerges in the large $N$ limit. We show that the critical gauge functions converges in the scaling limit to a limiting gauge function which is the unique solution to a dual Euler-Lagrange equation. This limiting gauge function determines the limit shape and vice versa. We further use our techniques to compute explicit limit shapes in some two and three dimensional examples, such as the Aztec diamond and "Aztec cuboid". This is one of the first stat mech models in dimensions $d\ge3$ where limit shapes can be computed explicitly.