Thick points for a Gaussian Free Field in 4 dimensions

Abstract

This article is concerned with the study of the fractal dimension of thick points for a 4-dimensional Gaussian Free Field. We adopt the definition of Gaussian Free Field on ${\mathbb{R}}^4$ introduced by Chen and Jakobson (2012) viewed as an abstract Wiener space with underlying Hilbert space $H^2({\mathbb{R}}^4)$. We can prove that for $0\le a\le 4$ the Hausdorff dimension of the set of $a$-high points is $4-a$. We also show that the set of thick points gives full mass to the 4-dimensional Liouville Quantum Gravity measure.

17 pages. Proof on the support of the Liouville Quantum Gravity added