Thick points for a Gaussian Free Field in 4 dimensions

TL;DR

This paper investigates the fractal geometry of thick points in a 4D Gaussian Free Field, establishing their Hausdorff dimension and connection to Liouville Quantum Gravity measures.

Contribution

It provides a precise calculation of the Hausdorff dimension of thick points in 4D GFF and links these points to Liouville Quantum Gravity.

Findings

Hausdorff dimension of $a$-high points is $4-a$ for $0 \\leq a \\leq 4$

Thick points carry full mass of the 4D Liouville Quantum Gravity measure

Extension of GFF analysis to four dimensions with explicit fractal 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 $\R^4$ introduced by Chen and Jakobson (2012) viewed as an abstract Wiener space with underlying Hilbert space $H^2(\R^4)$. We can prove that for $0\leq a \leq 4$ 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.