Gaussian free field and discrete Gaussian fluctuation conjecture for general periodic dimer models

Consider the height function on a large subgraph of the square lattice with $k \times \ell$ doubly periodic edge weights and general but sufficiently regular boundary conditions. Let the liquid region, with its induced complex structure, be denoted by $\tilde{\mathcal{R}}_0$, and let its double be the compact Riemann surface $\tilde{\mathcal{R}}$ of genus $\tilde{g}$. Let $\tilde{B}$ be the period matrix of $\tilde{\mathcal{R}}$, let $\tilde{f}_i:\tilde{\mathcal{R}}_0\to\mathbb{R}$ for $i=1,\dots,\tilde{g}$ be the associated harmonic functions, and let $(u_i,v_i)$ and $\tilde{H}_i$ be defined as in the source. Write $\mathfrak{g}_{\tilde{\mathcal{R}}_0}$ for a Gaussian free field on $\tilde{\mathcal{R}}_0$ with Dirichlet boundary conditions.

Gaussian free field–discrete Gaussian conjecture. The fluctuations of the dimer-model height function in the liquid region of a graph of scale $N$ are approximated in distribution by

$$\mathfrak{g}_{\tilde{\mathcal{R}}_0}(q)+\sum_{i=1}^{\tilde{g}}\tilde{f}_i(q)\tilde{Z}_i,$$

where $\mathfrak{g}_{\tilde{\mathcal{R}}_0}$ is independent of $\tilde{Z}=(\tilde{Z}_1,\dots,\tilde{Z}_{\tilde{g}})$, and $\tilde{Z}$ is distributed as a mean-subtracted discrete Gaussian with scale matrix $-\tilde{B}^{-1}$ and $N$-dependent shift parameter

$$N(\tilde{H}_1(u_1,v_1),\dots,\tilde{H}_{\tilde{g}}(u_{\tilde{g}},v_{\tilde{g}}))+e_0,$$

for some $e_0\in\mathbb{R}^{\tilde{g}}$.

This conjecture extends the Gaussian free field plus discrete Gaussian description of height fluctuations from the treated setting to general periodic dimer models and boundary conditions. The source gives no resolution status, so it is recorded as open.