Maximum of the Ginzburg.Landau fields

We study a two-dimensional massless field in a box with potential V(..(.)) 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 .(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.