Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Abstract

Given a measure ν on a regular planar domain D, the Gaussian multiplicative chaos measure of ν studied in this paper is the random measure ν obtained as the limit of the exponential of the γ-parameter circle averages of the Gaussian free field on D weighted by ν. We investigate the dimensional and geometric properties of these random measures. We first show that if ν is a finite Borel measure on D with exact dimension α > 0, then the associated GMC measure ν is non-degenerate and is almost surely exact dimensional with dimension α − γ 2 2 , provided γ 2 2 < α. We then show that if ν t is a Hölder-continuously parameterized family of measures then the total mass of ν t varies Hölder-continuously with t, provided that γ is sufficiently small. As an application we show that if γ < 0.28, then, almost surely, the orthogonal projections of the γ-Liouville quantum gravity measure µ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, µ has positive Fourier dimension almost surely.1991 Mathematics Subject Classification. 28A80, 60D05, 81T40.

On the action of multiplicative cascades on measures

“…This has been done for some infinitely divisible multiplicative chaos (starting with the log-normal case) acting on Lebesgue measure restricted to compact domains of R d [27,3,5], as well as for the Mandelbrot multiplicative cascades on the boundary of a regular tree when they act on the uniform measure [31], and more generally on so-called Markov measures [22] (see also [8]), as well as in the context of martingale convergence in branching random walks [10,35] . Results, not sharp but quite precise, also exist for general measures, in connection with estimates for their lower Hausdorff dimension [22,4,20].…”

Box-counting dimension in one-dimensional random geometry of multiplicative cascades

“…It originally arose in Mandelbrot's study of turbulence [20] but has since been investigated in its own right, see e.g. [3,4,5,6,13,15,17,21]. In one dimension the measure may be constructed iteratively by subdividing the unit line into dyadic intervals, multiplying the length of each subdivision by an i.i.d.…”

Box-Counting Dimension in One-Dimensional Random Geometry of Multiplicative Cascades

“…It originally arose in Mandelbrot's study of turbulence [22] but has since been investigated in its own right, see e.g. [3][4][5][6]14,17,19,23]. In one dimension the measure may be constructed iteratively by subdividing the unit line into dyadic intervals, multiplying the length of each subdivision by an i.i.d.…”