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.
- Balázs Bárány and Antti Käenmäki, Ledrappier-Young formula and exact dimensionality of self-affine measures, Adv. Math. 318 (2017), 88–129. MR 3689737, DOI https://doi.org/10.1016/j.aim.2017.07.015
- Julien Barral, Xiong Jin, and Benoît Mandelbrot, Uniform convergence for complex $[0,1]$-martingales, Ann. Appl. Probab. 20 (2010), no. 4, 1205–1218. MR 2676937, DOI https://doi.org/10.1214/09-AAP664
- Julien Barral and Benoît B. Mandelbrot, Multifractal products of cylindrical pulses, Probab. Theory Related Fields 124 (2002), no. 3, 409–430. MR 1939653, DOI https://doi.org/10.1007/s004400200220
- N. Berestycki. Introduction to the Gaussian Free Field and Liouville Quantum Gravity. Lecture notes, 2015.
- Nathanaël Berestycki, An elementary approach to Gaussian multiplicative chaos, Electron. Commun. Probab. 22 (2017), Paper No. 27, 12. MR 3652040, DOI https://doi.org/10.1214/17-ECP58
- N. Berestycki, S. Sheffield, and X. Sun, Equivalence of Liouville measure and Gaussian free field, arXiv:1410.5407 (2014).
- Bertrand Duplantier and Scott Sheffield, Liouville quantum gravity and KPZ, Invent. Math. 185 (2011), no. 2, 333–393. MR 2819163, DOI https://doi.org/10.1007/s00222-010-0308-1
- Fredrik Ekström, Tomas Persson, and Jörg Schmeling, On the Fourier dimension and a modification, J. Fractal Geom. 2 (2015), no. 3, 309–337. MR 3416842, DOI https://doi.org/10.4171/JFG/23
- Kenneth Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997. MR 1449135
- Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
- Kenneth Falconer, Jonathan Fraser, and Xiong Jin, Sixty years of fractal projections, Fractal geometry and stochastics V, Progr. Probab., vol. 70, Birkhäuser/Springer, Cham, 2015, pp. 3–25. MR 3558147, DOI https://doi.org/10.1007/978-3-319-18660-3_1
- Kenneth J. Falconer and Xiong Jin, Exact dimensionality and projections of random self-similar measures and sets, J. Lond. Math. Soc. (2) 90 (2014), no. 2, 388–412. MR 3263957, DOI https://doi.org/10.1112/jlms/jdu031
- Kenneth Falconer and Xiong Jin, Dimension conservation for self-similar sets and fractal percolation, Int. Math. Res. Not. IMRN 24 (2015), 13260–13289. MR 3436146, DOI https://doi.org/10.1093/imrn/rnv103
- De-Jun Feng and Huyi Hu, Dimension theory of iterated function systems, Comm. Pure Appl. Math. 62 (2009), no. 11, 1435–1500. MR 2560042, DOI https://doi.org/10.1002/cpa.20276
- Christophe Garban, Rémi Rhodes, and Vincent Vargas, Liouville Brownian motion, Ann. Probab. 44 (2016), no. 4, 3076–3110. MR 3531686, DOI https://doi.org/10.1214/15-AOP1042
- Michael Hochman and Pablo Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2) 175 (2012), no. 3, 1001–1059. MR 2912701, DOI https://doi.org/10.4007/annals.2012.175.3.1
- Xiaoyu Hu, Jason Miller, and Yuval Peres, Thick points of the Gaussian free field, Ann. Probab. 38 (2010), no. 2, 896–926. MR 2642894, DOI https://doi.org/10.1214/09-AOP498
- Xiaoyu Hu and S. James Taylor, Fractal properties of products and projections of measures in ${\bf R}^d$, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 527–544. MR 1269937, DOI https://doi.org/10.1017/S0305004100072285
- Brian R. Hunt and Vadim Yu. Kaloshin, How projections affect the dimension spectrum of fractal measures, Nonlinearity 10 (1997), no. 5, 1031–1046. MR 1473372, DOI https://doi.org/10.1088/0951-7715/10/5/002
- Jean-Pierre Kahane, Sur le chaos multiplicatif, Ann. Sci. Math. Québec 9 (1985), no. 2, 105–150 (French). MR 829798
- J.-P. Kahane and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot, Advances in Math. 22 (1976), no. 2, 131–145. MR 431355, DOI https://doi.org/10.1016/0001-8708%2876%2990151-1
- Jean-François Le Gall, Some properties of planar Brownian motion, École d’Été de Probabilités de Saint-Flour XX—1990, Lecture Notes in Math., vol. 1527, Springer, Berlin, 1992, pp. 111–235. MR 1229519, DOI https://doi.org/10.1007/BFb0084700
- B. B. Mandelbrot, A possible refinement of the lognormal hypothesis concerning the distribution of energy in intermittent turbulence, Statistical Models and Turbulence La Jolla, CA, Lecture Notes in Phys. no. 12, Springer, 1972, 333–351.
- B. B. Mandelbrot, Intermittent turbulence in self-similar cascades, divergence of high moments and dimension of the carrier, J. Fluid. Mech. 62 (1974), 331–358.
- J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4 (1954), 257–302. MR 63439, DOI https://doi.org/10.1112/plms/s3-4.1.257
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890
- Pertti Mattila, Fourier analysis and Hausdorff dimension, Cambridge Studies in Advanced Mathematics, vol. 150, Cambridge University Press, Cambridge, 2015. MR 3617376
- Jason Miller and Scott Sheffield, Imaginary geometry I: interacting SLEs, Probab. Theory Related Fields 164 (2016), no. 3-4, 553–705. MR 3477777, DOI https://doi.org/10.1007/s00440-016-0698-0
- Yuval Peres and Pablo Shmerkin, Resonance between Cantor sets, Ergodic Theory Dynam. Systems 29 (2009), no. 1, 201–221. MR 2470633, DOI https://doi.org/10.1017/S0143385708000369
- Jacques Peyrière, Turbulence et dimension de Hausdorff, C. R. Acad. Sci. Paris Sér. A 278 (1974), 567–569. MR 431354
- Rémi Rhodes and Vincent Vargas, Gaussian multiplicative chaos and applications: a review, Probab. Surv. 11 (2014), 315–392. MR 3274356, DOI https://doi.org/10.1214/13-PS218
- Haskell P. Rosenthal, On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273–303. MR 271721, DOI https://doi.org/10.1007/BF02771562
- Scott Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521–541. MR 2322706, DOI https://doi.org/10.1007/s00440-006-0050-1
- Scott Sheffield, Conformal weldings of random surfaces: SLE and the quantum gravity zipper, Ann. Probab. 44 (2016), no. 5, 3474–3545. MR 3551203, DOI https://doi.org/10.1214/15-AOP1055
- Pablo Shmerkin and Boris Solomyak, Absolute continuity of self-similar measures, their projections and convolutions, Trans. Amer. Math. Soc. 368 (2016), no. 7, 5125–5151. MR 3456174, DOI https://doi.org/10.1090/S0002-9947-2015-06696-3
- Pablo Shmerkin and Ville Suomala, Spatially independent martingales, intersections, and applications, Mem. Amer. Math. Soc. 251 (2018), no. 1195, v+102. MR 3756896, DOI https://doi.org/10.1090/memo/1195
- Bengt von Bahr and Carl-Gustav Esseen, Inequalities for the $r$th absolute moment of a sum of random variables, $1\leq r\leq 2$, Ann. Math. Statist. 36 (1965), 299–303. MR 170407, DOI https://doi.org/10.1214/aoms/1177700291
- W. Werner. Topics on the two-dimensional Gaussian Free Field. Lecture Notes from ETH Zürich, 2014.
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