Speaker: Matthew Nicoletti (UC Berkeley)

Title: The Doubly Periodic Aztec Diamond Dimer Model: Gaussian Free Field and Discrete Gaussians

Abstract:

Dimer models are random surface models whose scaling limits exhibit spatial phase separation. Due to its underlying integrability, the Aztec diamond dimer model in particular has been extensively researched in order to more precisely understand the various universal behaviors of dimer models. While global dimer model height fluctuations have been computed in many cases, until now the exact characterization of height fluctuations of a dimer model with gaseous facets appearing in the bulk has remained open.

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights, which lead to the formation of gaseous facets in the limit shape. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected rough region and a harmonic function with random, lattice-valued rough-gas boundary values. The boundary values are jointly distributed as a discrete Gaussian random vector. This discrete Gaussian distribution maintains a quasi-periodic dependence on~$N$, a phenomenon also observed in multi-cut random matrix models.
This is joint work with Tomas Berggren.