A Strong Central Limit Theorem for a Class of Random Surfaces
نویسندگان
چکیده
This paper is concerned with d = 2 dimensional lattice field models with action V (∇φ(·)), where V : Rd → 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〉A. In two dimensions g′′(x, μ) = ∂2g(x, μ)/∂μ2 is proportional to ln |x|. The main result of this paper is a bound on g′′′(x, μ) = ∂3g(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.
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