Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

TL;DR

This paper studies the geometric and dimensional properties of Gaussian multiplicative chaos measures derived from Gaussian free fields, establishing their exact dimensionality, stability under parameter variations, and projection properties in convex domains.

Contribution

It proves that GMC measures from measures with exact dimension are almost surely exact dimensional with a specific dimension shift, and analyzes their projection and Fourier properties.

Findings

GMC measures are almost surely exact dimensional with dimension $eta= ext{original dimension}-rac{ ext{gamma}^2}{2}$.

Total mass of parameterized GMC measures varies Hölder-continuously with the parameter.

Projections of GMC measures in convex domains are absolutely continuous with Hölder continuous densities for small gamma.

Abstract

Given a measure $\nu$ on a regular planar domain $D$, the Gaussian multiplicative chaos measure of $\nu$ studied in this paper is the random measure ${\widetilde \nu}$ obtained as the limit of the exponential of the $\gamma$-parameter circle averages of the Gaussian free field on $D$ weighted by $\nu$. We investigate the dimensional and geometric properties of these random measures. We first show that if $\nu$ is a finite Borel measure on $D$ with exact dimension $\alpha>0$, then the associated GMC measure ${\widetilde \nu}$ is non-degenerate and is almost surely exact dimensional with dimension $\alpha-\frac{\gamma^2}{2}$, provided $\frac{\gamma^2}{2}<\alpha$. We then show that if $\nu_t$ is a H\"{o}lder-continuously parameterized family of measures then the total mass of ${\widetilde \nu}_t$ varies H\"{o}lder-continuously with $t$, provided that $\gamma$ is sufficiently small. As an application we show that if $\gamma<0.28$, then, almost surely, the orthogonal projections of the $\gamma$-Liouville quantum gravity measure ${\widetilde \mu}$ on a rotund convex domain $D$ in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with H\"{o}lder continuous densities. Furthermore, ${\widetilde \mu}$ has positive Fourier dimension almost surely.