Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures


Authors: Kenneth Falconer and Xiong Jin
Journal: Trans. Amer. Math. Soc. 372 (2019), 2921-2957
MSC (2010): Primary 28A80, 60D05, 81T40
DOI: https://doi.org/10.1090/tran/7776
Published electronically: May 23, 2019
MathSciNet review: 3988598
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 nondegenerate 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 Hölder-continuously parameterized family of measures, then the total mass of ${\widetilde \nu }_t$ varies Hölder-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 Hölder continuous densities. Furthermore, ${\widetilde \mu }$ has positive Fourier dimension almost surely.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 28A80, 60D05, 81T40

Retrieve articles in all journals with MSC (2010): 28A80, 60D05, 81T40


Additional Information

Kenneth Falconer
Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland
MR Author ID: 65025
Email: kjf@st-andrews.ac.uk

Xiong Jin
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
MR Author ID: 901150
Email: xiong.jin@manchester.ac.uk

Keywords: Gaussian multiplicative chaos, absolute continuity, projection, dimension, Gaussian free field, circle average
Received by editor(s): August 22, 2017
Received by editor(s) in revised form: November 29, 2018
Published electronically: May 23, 2019
Article copyright: © Copyright 2019 American Mathematical Society