Title: The multinomial dimer model
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, introduced by Kenyon and Pohoata, studies a natural but non-uniform measure on $N$ dimer covers. While the standard dimer model ($N=1$) is exactly solvable only in two dimensions (i.e. on planar graphs), in the $N\to \infty$ limit, the multinomial dimer model turns out to be exactly solvable even in three (or higher) dimensions. In this talk, I will define the model and discuss new results in two and three dimensions, including: explicit formulas for the free energy, a large deviation principle, Euler-Lagrange equations for the limit shape and some explicit solutions, and possibly some preliminary results about fluctuations. This is joint work with Richard Kenyon.