A Strong Central Limit Theorem for a Class of Random Surfaces
A Strong Central Limit Theorem for a Class of Random Surfaces
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一类随机曲面的强中心极限定理
DOI：
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发表时间：
2011
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通讯作者：
T. Spencer
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作者：
J. Conlon;T. Spencer
This paper is concerned with d = 2 dimensional lattice field models with action $${V(\nabla\phi(\cdot))}$$V(∇ϕ(·)), where $${V : \mathbf{R}^d \rightarrow \mathbf{R}}$$V:Rd→R is a uniformly convex function. The fluctuations of the variable $${\phi(0) - \phi(x)}$$ϕ(0)-ϕ(x) are studied for large |x| via the generating function given by $${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$$g(x,μ)=ln〈eμ(ϕ(0)-ϕ(x))〉A. In two dimensions $${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$$g′′(x,μ)=∂2g(x,μ)/∂μ2 is proportional to $${\ln\vert x\vert}$$ln|x|. The main result of this paper is a bound on $${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$$g′′′(x,μ)=∂3g(x,μ)/∂μ3 which is uniform in $${\vert x \vert}$$|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.
This paper is concerned with d = 2 dimensional lattice field models with action $${V(\nabla\phi(\cdot))}$$V(∇ϕ(·)), where $${V : \mathbf{R}^d \rightarrow \mathbf{R}}$$V:Rd→R is a uniformly convex function. The fluctuations of the variable $${\phi(0) - \phi(x)}$$ϕ(0)-ϕ(x) are studied for large |x| via the generating function given by $${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$$g(x,μ)=ln〈eμ(ϕ(0)-ϕ(x))〉A. In two dimensions $${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$$g′′(x,μ)=∂2g(x,μ)/∂μ2 is proportional to $${\ln\vert x\vert}$$ln|x|. The main result of this paper is a bound on $${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$$g′′′(x,μ)=∂3g(x,μ)/∂μ3 which is uniform in $${\vert x \vert}$$|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.