Thick points for a Gaussian Free Field in 4 dimensions

Abstract

This article is concerned with the study of fractal properties of thick points for a 4-dimensional Gaussian Free Field. We adopt the definition of Gaussian Free Field on R4 introduced by Chen and Jakobson (2012) viewed as an abstract Wiener space with underlying Hilbert space H2(R4). We can prove that for 0≤a≤4, the Hausdorff dimension of the set of a-high points is 4−a. We also show that the thick points give full mass to the Liouville Quantum Gravity measure on R4.