Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models

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.