Thick points of the Gaussian free field

TL;DR

This paper characterizes the geometric structure of thick points in the Gaussian free field, establishing their Hausdorff dimension, measure properties, and conformal invariance, linking to Liouville quantum gravity.

Contribution

It provides a detailed analysis of the Hausdorff dimension and measure of thick points in the Gaussian free field, and demonstrates their conformal invariance, connecting to quantum gravity models.

Findings

Hausdorff dimension of T(a;U) is 2-a for 0≤a≤2

T(a;U) has infinite Hausdorff measure for 0<a≤2

T(a;U) is empty for a>2

Abstract

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 $0<a\leq2$ and $\nu_2(T(0;U))=\nu_2(U)$ almost surely, where $\nu_{\alpha}$ is the Hausdorff-$\alpha$ measure, and that $T(a;U)$ is almost surely empty when $a>2$. 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.