Gaussian Free Field and Discrete Gaussians in Periodic Dimer 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.