The multinomial dimer model

An N-dimer cover of a graph is a collection of edges (with multiplicity) such that each vertex is contained in exactly N edges in the collection. The multinomial dimer model is a natural probability measure on N-dimer covers. We study the behavior of these measures on periodic bipartite graphs in R d , in the scaling limit as the multiplicity N and then the size of the graph go to infinity. In this iterated limit, we prove a large deviation principle, where the rate function is the integral of an explicit surface tension, and show that random configurations concentrate on a limit shape which is the unique solution to an associated Euler-Lagrange equation. We further show that the associated critical gauge functions, which exist in the N → ∞ limit on each finite graph, converge in the scaling limit to a limiting gauge function which solves a dual Euler-Lagrange equation. We use our techniques to compute explicit limit shapes in some two and three dimensional examples, such as the Aztec diamond and “Aztec cuboid”. These 3d examples are the first stat mech models in dimensions d ≥ 3 where limit shapes can be computed explicitly.