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Transactions of the American Mathematical Society

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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
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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.


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Additional Information

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

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

DOI: https://doi.org/10.1090/tran/7776
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