Harmonic Pinnacles in the Discrete Gaussian Model

Abstract

The 2D Discrete Gaussian model gives each height function ${\eta : {\mathbb{Z}^2\to\mathbb{Z}}}$ a probability proportional to ${\exp(-\beta \mathcal{H}(\eta))}$, where ${\beta}$ is the inverse-temperature and ${\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}$ sums over nearest-neighbor bonds. We consider the model at large fixed ${\beta}$, where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an ${L\times L}$ box with 0 boundary conditions concentrates on two integers M, M + 1 with ${M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}$. The key is a large deviation estimate for the height at the origin in ${\mathbb{Z}^{2}}$, dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on ${\eta\geq 0}$ (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where ${H\sim M/\sqrt{2}}$. 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 ${\sqrt{\log L}}$. Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.