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 √a/π log 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 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 Γ (dz) = e √2πγF(z) dz considered by Duplantier and Sheffield.