A strong central limit theorem for a class of random surfaces

TL;DR

This paper establishes a strong central limit theorem for a class of two-dimensional lattice field models with convex interactions, providing bounds on higher-order derivatives of the generating function that describe fluctuations.

Contribution

It proves a uniform bound on the third derivative of the generating function for a class of convex potentials, advancing understanding of fluctuation behavior in 2D lattice field models.

Findings

Bound on the third derivative of the generating function is uniform in |x|.

Fluctuations of the field difference grow logarithmically with distance.

Method employs advanced integration by parts and singular integral estimates.

Abstract

This paper is concerned with $d=2$ dimensional lattice field models with action $V(\na\phi(\cdot))$, where $V:\R^d\ra \R$ is a uniformly convex function. The fluctuations of the variable $\phi(0)-\phi(x)$ are studied for large $|x|$ via the generating function given by $g(x,\mu) = \ln <e^{\mu(\phi(0) - \phi(x))}>_{A}$. In two dimensions $g"(x,\mu)=\pa^2g(x,\mu)/\pa\mu^2$ is proportional to $\ln|x|$. The main result of this paper is a bound on $g"'(x,\mu)=\pa^3 g(x,\mu)/\pa \mu^3$ which is uniform in $|x|$ for a class of convex $V$. The proof uses integration by parts following Helffer-Sj\"{o}strand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.