Thick points of the Gaussian free field

Original · EN

Let U 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/π1/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 Γ(dz)=e√2πγF(z)dz considered by Duplantier and Sheffield.