Source: Annals of Probability, Volume 48, Number 6, 2647--2679.
Abstract:
We study a two-dimensional massless field in a box with potential $V(\nabla \phi (\cdot ))$ and zero boundary condition, where $V$ is any symmetric and uniformly convex function. Naddaf–Spencer ( Comm. Math. Phys. 183 (1997) 55–84) and Miller ( Comm. Math. Phys. 308 (2011) 591–639) proved that the rescaled macroscopic averages of this field converge to a continuum Gaussian free field. In this paper, we prove that the distribution of local marginal $\phi (x)$, for any $x$ in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and the dimension of high points of this field, thus generalizing the results of Bolthausen–Deuschel–Giacomin ( Ann. Probab. 29 (2001) 1670–1692) and Daviaud ( Ann. Probab. 34 (2006) 962–986) for the discrete Gaussian free field.
