UW Probability Seminar

# UW Probability Seminar

Time: Monday, December 2, 2013 at 2:30 pm.

Location: MEB 243

Speaker: Eyal Lubetzky (Microsoft Research)

Title: Harmonic pinnacles in the Discrete Gaussian model

Abstract: The 2D Discrete Gaussian model gives each height function $\eta: Z^2 \to Z$ a probability proportional to $\exp[-\beta H(\eta)]$, where $\beta$ is the inverse-temperature and $H (\eta) = \sum (\eta_x-\eta_y)^2$ sums over nearest-neighbor bonds $(x,y)$. 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 $M,M+1$ where $M\sim [(2/\pi\beta)\log L\log\log L]^{1/2}$. The key is a large deviation estimate for the height at the origin in $Z^2$, 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 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, El-Mellouki and Frohlich (1986). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to $p$-harmonic analysis and alternating sign matrices.

Joint work with Fabio Martinelli and Allan Sly.

Archive of previous talks

The University of Washington is committed to providing access, equal opportunity and reasonable accommodation in its services, programs, activities, education and employment for individuals with disabilities. To request disability accommodation contact the Disability Services Office at least ten days in advance at: 206-543-6450 (voice), 206-543-6452 (TTY), 206-685-7264 (FAX), or dso@u.washington.edu