2025/06/13 by Kenyon, Richard, Wolfram, Catherine
#60F10 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2506.12171
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 \mathbb Rd, 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.