Skip to content

Research at St Andrews

Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

Research output: Contribution to journalArticle

DOI

Open Access permissions

Open

Author(s)

Kenneth Falconer, Xiong Jin

School/Research organisations

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 non-degenerate 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.
Close

Details

Original languageEnglish
Pages (from-to)2921-2957
Number of pages37
JournalTransactions of the American Mathematical Society
Volume372
Issue number4
Early online date23 May 2019
DOIs
Publication statusE-pub ahead of print - 23 May 2019

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Marstrand's Theorem revisited: projecting sets of dimension zero

    Beresnevich, V., Falconer, K., Velani, S. & Zafeiropoulos, A., 15 Apr 2019, In : Journal of Mathematical Analysis and Applications. 472, 2, p. 1820-1845 26 p.

    Research output: Contribution to journalArticle

  2. A capacity approach to box and packing dimensions of projections of sets and exceptional directions

    Falconer, K. J., 12 Mar 2019, (Accepted/In press) In : Journal of Fractal Geometry. 20 p.

    Research output: Contribution to journalArticle

  3. Self-stabilizing processes based on random signs

    Falconer, K. J. & Lévy Véhel, J., 29 Sep 2018, In : Journal of Theoretical Probability. First Online, 16 p.

    Research output: Contribution to journalArticle

  4. Planar self-affine sets with equal Hausdorff, box and affinity dimensions

    Falconer, K. & Kempton, T., Jun 2018, In : Ergodic Theory and Dynamical Systems. 38, 4, p. 1369-1388 20 p.

    Research output: Contribution to journalArticle

  5. Self-stabilizing processes

    Falconer, K. J. & Lévy Vehel, J., 2018, In : Stochastic Models. 34, 4, p. 409-434 26 p.

    Research output: Contribution to journalArticle

Related by journal

  1. Orbits of primitive k-homogenous groups on (n-k)-partitions with applications to semigroups

    Araújo, J., Bentz, W. & Cameron, P. J., 1 Jan 2019, In : Transactions of the American Mathematical Society. 371, 1, p. 105-136

    Research output: Contribution to journalArticle

  2. Regularity of Kleinian limit sets and Patterson-Sullivan measures

    Fraser, J. M., 21 Jun 2019, In : Transactions of the American Mathematical Society. Early View, 33 p.

    Research output: Contribution to journalArticle

  3. Generating sets of finite groups

    Cameron, P. J., Lucchini, A. & Roney-Dougal, C. M., 4 Apr 2018, In : Transactions of the American Mathematical Society. 370, 9, p. 6751-6770

    Research output: Contribution to journalArticle

  4. Some undecidability results for asynchronous transducers and the Brin-Thompson group 2V

    Belk, J. & Bleak, C., May 2017, In : Transactions of the American Mathematical Society. 369, 5, p. 3157-3172 16 p.

    Research output: Contribution to journalArticle

ID: 240125030