Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models

TL;DR

This paper studies height fluctuations in periodic dimer models, revealing they approximate a combination of a Gaussian free field and a harmonic function with discrete Gaussian boundary conditions, highlighting complex boundary dependencies.

Contribution

It demonstrates that height fluctuations in periodic dimer models can be decomposed into independent Gaussian free field and harmonic components with discrete Gaussian boundary conditions, a novel insight.

Findings

Height fluctuations approximate a Gaussian free field plus a harmonic function.

Boundary values follow a discrete Gaussian distribution with quasi-periodic dependence.

The discrete Gaussian distribution exhibits phenomena similar to multi-cut random matrix models.

Abstract

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected liquid region and a harmonic function with random liquid-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.