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 ${\widetilde \nu}$ 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.