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 $0\le a\le 2$ the Hausdorff dimension of $T(a;U)$ is almost surely $2-a$, that ${\nu }_{2-a}(T(a;U))=\infty$ when $0<a\le 2$ 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.

Published in at http://dx.doi.org/10.1214/09-AOP498 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Extremes of the discrete two-dimensional Gaussian free field

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