  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.
