Exact dimensionality and projection properties of Gaussian multiplicative chaos measures
View/ Open
Date
15/08/2019Metadata
Show full item recordAltmetrics Handle Statistics
Altmetrics DOI Statistics
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.
Citation
Falconer , K & Jin , X 2019 , ' Exact dimensionality and projection properties of Gaussian multiplicative chaos measures ' , Transactions of the American Mathematical Society , vol. 372 , no. 4 , pp. 2921-2957 . https://doi.org/10.1090/tran/7776
Publication
Transactions of the American Mathematical Society
Status
Peer reviewed
ISSN
0002-9947Type
Journal article
Rights
Copyright © 2018, American Mathematical Society. This work has been made available online in accordance with the publisher's policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1090/tran/7776
Description
Paper originally entitled 'Hölder continuity of the Liouville Quantum Gravity measure'Collections
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.