Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

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 $\tilde{\nu }$ 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 $\tilde{\nu }$ 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 Hölder-continuously parameterized family of measures then the total mass of ${\tilde{\nu }}_t$ varies Hölder-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 $\tilde{\mu }$ on a rotund convex domain $D$ in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, $\tilde{\mu }$ has positive Fourier dimension almost surely.

arXiv comment33 pages, 1 figure