Thick points of the Gaussian free field

Abstract

Let $U\subseteq\mathbf{C}$ 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 $\int_U\nabla f(x)\cdot \nabla g(x)\,dx$. The set $T(a;U)$ of $a$-thick points of $F$ consists of those $z\in U$ 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 $r\to 0$. We show that for each $0\leq a\leq2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that $\nu_{2-a}(T(a;U))=\infty$ when $0<a\leq2$ and $\nu_2(T(0;U))=\nu_2(U)$ almost surely, where $\nu_{\alpha}$ is the Hausdorff-$\alpha$ 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 $\gamma$ given formally by $\Gamma(dz)=e^{\sqrt{2\pi}\gamma F(z)}\,dz$ considered by Duplantier and Sheffield.