Local central limit theorem for gradient field models

We consider the gradient field model in \(\left[ -N,N\right] ^{2}\cap {\mathbb {Z}}^{2}\) ${\left[-,N,,,N\right]}^2\cap {\mathbb{Z}}^2$ with a uniformly convex interaction potential. Naddaf–Spencer (Comm Math Phys 183(1):55–84, 1997) and Miller (Comm Math Phys 908(3):591–639, 2011) 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 \(\phi (0)/\sqrt{\log N}\) $\varphi (0)/\sqrt{logN}$ converges uniformly in \({\mathbb {R}}\) $\mathbb{R}$ to a Gaussian density, with a Berry-Esseen type bound. This implies the distribution of \(\phi (0)\) $\varphi (0)$ is sufficiently ‘Gaussian like’ between \([-\sqrt{\log N}, \sqrt{\log N}]\) $[-\sqrt{logN},\sqrt{logN}]$ .