Local central limit theorem for gradient field models

TL;DR

This paper establishes a local central limit theorem for the gradient field model with convex potential, showing the distribution of the field at the origin approximates a Gaussian with explicit bounds, enhancing understanding of its probabilistic behavior.

Contribution

It proves a uniform Gaussian approximation for the distribution of the field at the origin, with a Berry-Esseen bound, extending previous convergence results to a local limit theorem.

Findings

Distribution of (0)/\u221a{\u2212}log N converges to Gaussian density

Distribution of (0) is approximately Gaussian within  range

Provides a Berry-Esseen type bound for the convergence

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 $\phi(0)/\sqrt{\log N}$ converges uniformly to a Gaussian density, with a Berry-Esseen type bound. This implies the distribution of $\phi(0)$ is sufficiently `Gaussian like' between $[-\sqrt {\log N}, \sqrt {\log N}]$.