Harmonic pinnacles in the Discrete Gaussian model

The 2D Discrete Gaussian model gives each height function $\eta : $\mathbb{Z}$^2\to$\mathbb{Z}$$aprobabilityproportionalto$\exp(-\beta $\mathcal{H}$(\eta))$,where$\beta$istheinverse-temperatureand$$\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 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 $\eta \ge 0$ (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels $H,H+1$ where $H\sim M\mathrm{/}\sqrt{2}$. This in particular pins down the asymptotics, and corrects the order, in results of Bricmont, El-Mellouki and Fr"ohlich (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.