A strong central limit theorem for a class of random surfaces

Abstract

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

14 pages