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March 2010 Thick points of the Gaussian free field
Xiaoyu Hu, Jason Miller, Yuval Peres
Ann. Probab. 38(2): 896-926 (March 2010). DOI: 10.1214/09-AOP498

Abstract

Let UC be a bounded domain with smooth boundary and let F be an instance of the continuum Gaussian free field on U with respect to the Dirichlet inner product Uf(x)⋅∇g(x) dx. The set T(a; U) of a-thick points of F consists of those zU such that the average of F on a disk of radius r centered at z has growth $\sqrt{a/\pi}\log\frac{1}{r}$ as r→0. We show that for each 0≤a≤2 the Hausdorff dimension of T(a; U) is almost surely 2−a, that ν2−a(T(a; U))=∞ when 0<a≤2 and ν2(T(0; U))=ν2(U) almost surely, where να is the Hausdorff-α measure, and that T(a; U) is almost surely empty when a>2. Furthermore, we prove that T(a; U) is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter γ given formally by $\Gamma(dz)=e^{\sqrt{2\pi}\gamma F(z)}\,dz$ considered by Duplantier and Sheffield.

Citation

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Xiaoyu Hu. Jason Miller. Yuval Peres. "Thick points of the Gaussian free field." Ann. Probab. 38 (2) 896 - 926, March 2010. https://doi.org/10.1214/09-AOP498

Information

Published: March 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1201.60047
MathSciNet: MR2642894
Digital Object Identifier: 10.1214/09-AOP498

Subjects:
Primary: 60G15, 60G18, 60G60

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 2 • March 2010
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