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Almost sure convergence and second moments of geometric functionals of fractal percolation

Part of: Geometric probability and stochastic geometry Equilibrium statistical mechanics Special processes Classical measure theory

Published online by Cambridge University Press:  28 November 2023

Michael A. Klatt  and
Steffen Winter
Michael A. Klatt*
Affiliation:
Heinrich-Heine-Universität Düsseldorf
Steffen Winter*
Affiliation:
Karlsruhe Institute of Technology
*
*Postal address: Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany; Experimental Physics, Saarland University, Center for Biophysics, 66123 Saarbrücken, Germany. Present address: Institut für KI Sicherheit, Deutsches Zentrum für Luft- und Raumfahrt, Wilhelm-Runge-Straße 10, 89081 Ulm, Germany; Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt, 51170 Köln, Germany; Department of Physics, Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 Munich, Germany. Email address: michael.klatt@dlr.de
**Postal address: Karlsruhe Institute of Technology, Department of Mathematics, Englerstr. 2, 76128 Karlsruhe, Germany. Email address: steffen.winter@kit.edu
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Abstract

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in ${\mathbb R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows one, in particular, to determine their expectations and covariance structure. We also show the convergence of the rescaled expectations and variances of the intrinsic volumes of the construction steps to the expectations and variances of the limit variables, and we give rates for this convergence in some cases. These results significantly extend our previous work, which addressed only limits of expectations of intrinsic volumes.

Keywords

Fractal percolationMandelbrot percolationMinkowski functionalsintrinsic volumescurvature measuresfractal curvaturesrandom self-similar setrenewal theoremGalton–Watson processbranching random walk

MSC classification

Primary: 28A80: Fractals
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 3 , September 2024 , pp. 927 - 959
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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