Abstract
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.
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Conlon, J.G., Spencer, T. A Strong Central Limit Theorem for a Class of Random Surfaces. Commun. Math. Phys. 325, 1–15 (2014). https://doi.org/10.1007/s00220-013-1843-6
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DOI: https://doi.org/10.1007/s00220-013-1843-6