Two-periodic weighted dominos and the sine-Gordon field at the free fermion point: I

TL;DR

This paper demonstrates that the height field of a two-periodic dimer model at a smooth-rough transition converges to the massless sine-Gordon field at the free fermion point, revealing a connection between tiling models and non-Gaussian fields.

Contribution

It provides a rigorous asymptotic analysis linking the dimer model's height field to the sine-Gordon field at a critical transition, with explicit correlation functions.

Findings

Correlation functions converge to those of the sine-Gordon field

The limiting field interpolates between Gaussian free field and white noise

Explicit connection established between tiling models and non-Gaussian fields

Abstract

In this paper we investigate the height field of a dimer model/random domino tiling on the plane at a smooth-rough (i.e. gas-liquid) transition. We prove that the height field at this transition has two-point correlation functions which limit to those of the massless sine-Gordon field at the free fermion point, with parameters $(4\pi, z)$ where $z\in \mathbb{R}\setminus \{0\}$. The dimer model is on $\epsilon \mathbb{Z}^2$ and has a two-periodic weight structure with weights equal to either 1 or $a=1-C|z|\epsilon$, for $0<\epsilon$ small (tending to zero). In order to obtain this result, we provide a direct asymptotic analysis of a double contour integral formula of the correlation kernel of the dimer model found by Fourier analysis. The limiting field interpolates between the Gaussian free field and white noise and the main result gives an explicit connection between tiling/dimer models and the law of a two-dimensional non-Gaussian field.