A Strong Central Limit Theorem for a Class of Random Surfaces

This paper is concerned with d = 2 dimensional lattice field models with action ${V(\nabla\phi(\cdot))}$ , where ${V : \mathbf{R}^d \rightarrow \mathbf{R}}$ 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 ${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$ . In two dimensions ${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$ is proportional to ${\ln\vert x\vert}$ . The main result of this paper is a bound on ${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$ which is uniform in ${\vert x \vert}$ 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.