Paper detail

A strong central limit theorem for a class of random surfaces

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 $ϕ(0)-ϕ(x)$ are studied for large $|x|$ via the generating function given by $g(x,μ) = \ln <e^{μ(ϕ(0) - ϕ(x))}>_{A}$. 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.