A strong central limit theorem for a class of random surfaces

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This paper is concerned with d=2 dimensional lattice field models with action V(ϕ(·)), where V:ᵈ 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) = <eμ⁽ϕ⁽⁰⁾ ⁻ ϕ⁽ˣ⁾⁾>ₐ. In two dimensions g"(x,μ)=²g(x,μ)/μ² is proportional to |x|. The main result of this paper is a bound on g"'(x,μ)=³ g(x,μ)/ μ³ 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.