The Annals of Probability
- Ann. Probab.
- Volume 38, Number 2 (2010), 896-926.
Thick points of the Gaussian free field
Let U⊆C 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 ∫U∇f(x)⋅∇g(x) dx. The set T(a; U) of a-thick points of F consists of those z∈U such that the average of F on a disk of radius r centered at z has growth 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 considered by Duplantier and Sheffield.
Ann. Probab., Volume 38, Number 2 (2010), 896-926.
First available in Project Euclid: 9 March 2010
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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. https://projecteuclid.org/euclid.aop/1268143535