Domino tilings of black-and-white Temperleyan cylinders

TL;DR

This paper proves that height fluctuations in the dimer model on cylindrical domains with specific boundary conditions converge to a Gaussian Free Field plus an independent Gaussian component, revealing new insights into the model's limiting behavior.

Contribution

It establishes the convergence of height fluctuations to a Gaussian Free Field in cylindrical domains with Temperleyan boundaries, including the structure of the limit and the role of the discrete Gaussian distribution.

Findings

Height fluctuations converge to Gaussian Free Field plus an independent Gaussian component.

Limit of dimer coupling functions is holomorphic but not conformally covariant.

Discrete Gaussian distribution naturally appears in the doubly connected setup.

Abstract

We consider the dimer model in cylindrical domains $\Omega_\delta$ on square grids of mesh size $\delta$ with two Temperleyan boundary components of different colors. Assuming that the $\Omega_\delta$ approximate a cylindrical domain $\Omega$ as $\delta\to 0$, we prove the convergence of height fluctuations to the Gaussian Free Field in $\Omega$ plus an independent discrete Gaussian multiple of the harmonic measure of one of the boundary components. The limit of the dimer coupling functions on $\Omega_\delta$ is holomorphic in $\Omega$ but not conformally covariant. Given this, we determine the limiting structure of height fluctuations from general principles rather than from explicit computations. In particular, our analysis justifies the inevitable appearance of the discrete Gaussian distribution in the doubly connected setup.