Maximum of the Ginzburg-Landau fields

We study two dimensional massless field in a box with potential $V\left( \nabla \phi \left( \cdot \right) \right)$ and zero boundary condition, where $V$ is any symmetric and uniformly convex function. Naddaf-Spencer and Miller proved the macroscopic averages of this field converge to a continuum Gaussian free field. In this paper we prove the distribution of local marginal $\phi \left( x\right)$, for any $x$ in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and dimension of high points of this field, thus generalize the results of Bolthausen-Deuschel-Giacomin and Daviaud for the discrete Gaussian free field.