Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Kenneth Falconer, Xiong Jin

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
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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 nondegenerate and is almost surely exact dimensional with dimension α-γ2/2, provided γ2/2<α. We then show that if νt is a Hölder-continuously parameterized family of measures, then the total mass of ^νt^ 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 ^ν^ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, ^ν^ has positive Fourier dimension almost surely.
Original languageEnglish
Pages (from-to)2921-2957
Number of pages37
JournalTransactions of the American Mathematical Society
Volume372
Issue number4
Early online date23 May 2019
DOIs
Publication statusPublished - 15 Aug 2019

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