Thick points of the Gaussian free field

Abstract

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 $\sqrt{a/\pi}\log\frac{1}{r}$ as r→0. We show that for each 0≤a≤2 the Hausdorff dimension of T(a; U) is almost surely 2−a, that ν_2−a(T(a; U))=∞ when 0<a≤2 and ν₂(T(0; U))=ν₂(U) almost surely, where ν_α 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.