Thick points of the Gaussian free field

Details Thick points of the Gaussian free field Thick points of the Gaussian free field

2009-02-23

arxiv.org/abs/0902.3842v3

Annals of Probability 2010, Vol. 38, No. 2, 896-926

Mathematics

Probability

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

Scientific paper

10.1214/09-AOP498

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 $02$. 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.

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