A strong central limit theorem for a class of random surfaces

Details A strong central limit theorem for a class of random surfaces A strong central limit theorem for a class of random surfaces

2011-05-13

arxiv.org/abs/1105.2814v2

Physics

Mathematical Physics

15 pages

Scientific paper

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 $g(x,\mu) = \ln _{A}$. 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.

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