The Annals of Probability

Thick points of the Gaussian free field

Xiaoyu Hu, Jason Miller, and Yuval Peres

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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.

Article information

Ann. Probab., Volume 38, Number 2 (2010), 896-926.

First available in Project Euclid: 9 March 2010

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Zentralblatt MATH identifier

Primary: 60G60: Random fields 60G15: Gaussian processes 60G18: Self-similar processes

Gaussian free field thick points extremal points Hausdorff dimension fractal conformal invariance


Hu, Xiaoyu; Miller, Jason; Peres, Yuval. Thick points of the Gaussian free field. Ann. Probab. 38 (2010), no. 2, 896--926. doi:10.1214/09-AOP498.

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