Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Abstract

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