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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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
Published electronically: May 23, 2019
MathSciNet review: 3988598
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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.

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Additional Information

Kenneth Falconer
Affiliation: Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland
MR Author ID: 65025

Xiong Jin
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
MR Author ID: 901150

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