A strong central limit theorem for a class of random surfaces

This paper is concerned with $d=2$ dimensional lattice field models with action $V(\na\phi(\cdot))$, 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.