Local central limit theorem for gradient field models

Abstract

We consider the gradient field model in ${\left[-N,N\right]}^2\cap {\mathbb{Z}}^2$ with a uniformly convex interaction potential. Naddaf-Spencer \cite{NS} and Miller \cite{Mi} proved that the macroscopic averages of linear statistics of the field converge to a continuum Gaussian free field. In this paper we prove the distribution of $\varphi (0)/\sqrt{\log N}$ converges uniformly to a Gaussian density, with a Berry-Esseen type bound. This implies the distribution of $\varphi (0)$ is sufficiently `Gaussian like' between $[-\sqrt{\log N},\sqrt{\log N}]$.