Harmonic Pinnacles in the Discrete Gaussian Model

Abstract

The 2D Discrete Gaussian model gives each height function a probability proportional to , where is the inverse-temperature and sums over nearest-neighbor bonds. We consider the model at large fixed , where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an box with 0 boundary conditions concentrates on two integers M, M + 1 with . The key is a large deviation estimate for the height at the origin in , dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where . This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5-6):743-798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order . Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.