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Thick points of the Gaussian free field. (English) Zbl 1201.60047
Let \(U\subseteq \mathbb{C}\) 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 \(\int_{U}\nabla f\left( x\right) \cdot \nabla g\left( x\right) \mathrm{d}x\). The set \( T\left( a;U\right) \) 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 \(\sqrt{a/\pi }\log \displaystyle\frac{1}{r}\) as\(\;r\rightarrow 0\). The authors show that for each \(0\leq a\leq 2\) the Hausdorff dimension of \( T\left( a;U\right) \) is almost surely \(2-a,\) that \(v_{2-a}\left( T\left( a;U\right) \right) =\infty \) when \(0<a\leq 2\) and \(\nu _{2}\left( T\left( 0;U\right) \right) =\nu _{2}\left( U\right) \) almost surely, where \(\nu _{\alpha }\) is a Hausdorff-\(\alpha \) measure, and that \(T\left( a;U\right) \) is almost surely empty when \(a>2\). Furthermore, they prove that \(T\left( a;U\right) \) 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 \(\Gamma \left( \mathrm{d}z\right) =e^{\sqrt{2\pi }\gamma F\left( z\right) }\mathrm{d}z\) considered by B. Duplantier and S. Sheffield [arXiv:0808.1560].

60G60 Random fields
60G15 Gaussian processes
60G18 Self-similar stochastic processes
28A80 Fractals
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