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Geometric functionals of fractal percolation

Part of: Classical measure theory Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  03 December 2020

Michael A. Klatt  and
Steffen Winter
Michael A. Klatt*
Affiliation:
Princeton University
Steffen Winter*
Affiliation:
Karlsruhe Institute of Technology
*
*Postal address: Department of Physics, Princeton University, Princeton, New Jersey 08544, USA. Email: mklatt@princeton.edu
**Postal address: Karlsruhe Institute of Technology, Department of Mathematics, 76128 Karlsruhe, Germany. Email: steffen.winter@kit.edu
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Abstract

Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system-spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process F. They arise as limits of expected functionals of finite approximations of F. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.

Keywords

Fractal percolationMandelbrot percolationMinkowski functionalsintrinsic volumescurvature measuresfractal curvaturesrandom self-similar setpercolation threshold

MSC classification

Primary: 28A80: Fractals
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1085 - 1126
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Copyright
© Applied Probability Trust 2020

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References

Bobrowski, O. and Skraba, P. (2020). Homological percolation and the Euler characteristic. Phys. Rev. E. 101, 16 pp. Available at https://link.aps.org/doi/10.1103/PhysRevE.101.032304.CrossRefGoogle Scholar
Broman, E. I. and Camia, F. (2008). Large-N limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in Mandelbrot’s fractal percolation process. Electron. J. Prob. 13, 980–999.10.1214/EJP.v13-511CrossRefGoogle Scholar
Broman, E. I. and Camia, F. (2010). Universal behavior of connectivity properties in fractal percolation models. Electron. J. Prob. 15, 1394–1414.10.1214/EJP.v15-805CrossRefGoogle Scholar
Broman, E. I., Camia, F., Joosten, M. and Meester, R. (2013). Dimension (in)equalities and Hölder continuous curves in fractal percolation. J. Theoret. Prob. 26, 836–854.CrossRefGoogle Scholar
Chayes, J. T. and Chayes, L. (1989). The large-N limit of the threshold values in Mandelbrot’s fractal percolation process. J. Phys. A 22, L501–L506.10.1088/0305-4470/22/11/009CrossRefGoogle Scholar
Chayes, J. T., Chayes, L. and Durrett, R. (1988). Connectivity properties of Mandelbrot’s percolation process. Prob. Theory Relat. Fields 77, 307–324.CrossRefGoogle Scholar
Dekking, F. M. and Meester, R. W. J. (1990). On the structure of Mandelbrot’s percolation process and other random Cantor sets. J. Statist. Phys. 58, 1109–1126.10.1007/BF01026566CrossRefGoogle Scholar
Don, H. (2015). New methods to bound the critical probability in fractal percolation. Random Structures Algorithms 47, 710–730.CrossRefGoogle Scholar
Falconer, K. J. (1986). Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559–582.CrossRefGoogle Scholar
Göring, D., Klatt, M. A., Stegmann, C. and Mecke, K. (2013). Morphometric analysis in gamma-ray astronomy using Minkowski functionals: source detection via structure quantification. Astron. Astrophys. 555, A38.10.1051/0004-6361/201321136CrossRefGoogle Scholar
Graf, S. (1987). Statistically self-similar fractals. Prob. Theory Relat. Fields 74, 357–392.CrossRefGoogle Scholar
Klatt, M. A., Schröder-Turk, G. E. and Mecke, K. (2017). Anisotropy in finite continuum percolation: threshold estimation by Minkowski functionals. J. Statist. Mech. Theory Exp. 2017, 023302.CrossRefGoogle Scholar
Klatt, M. A. and Winter, S. Geometric functionals of fractal percolation. II. Almost sure convergence and second moments. In preparation.Google Scholar
Klatt, M. A. and Winter, S. (2019). FractalPercolationMink—Approximate fractal percolation and compute its Euler characteristic. GitHub repository. Available at http://github.com/michael-klatt/fractal- percolation-mink.Google Scholar
Knüfing, L., Schollmeyer, H., Riegler, H. and Mecke, K. (2005). Fractal analysis methods for solid alkane monolayer domains at SiO2/air interfaces. Langmuir 21, 992–1000.10.1021/la0476783CrossRefGoogle ScholarPubMed
Mandelbrot, B. B. (1974). Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358.CrossRefGoogle Scholar
Matsumoto, M. and Nishimura, T. (1998). Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Trans. Model. Comput. Simul. 8, 3–30.CrossRefGoogle Scholar
Mauldin, R. D. and Williams, S. C. (1986). Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.CrossRefGoogle Scholar
Mecke, K. R. and Seyfried, A. (2002). Strong dependence of percolation thresholds on polydispersity. EPL – Europhys. Lett. 58, 28–34.CrossRefGoogle Scholar
Mecke, K. R. and Wagner, H. (1991). Euler characteristic and related measures for random geometric sets. J. Statist. Phys. 64, 843–850.CrossRefGoogle Scholar
Neher, R. A., Mecke, K. and Wagner, H. (2008). Topological estimation of percolation thresholds. J. Statist. Mech. Theory Exp. 2008, P01011.CrossRefGoogle Scholar
Roubin, E. and Colliat, J.-B. (2016). Critical probability of percolation over bounded region in n-dimensional Euclidean space. J. Statist. Mech. Theory Exp. 2016, 033306.10.1088/1742-5468/2016/03/033306CrossRefGoogle Scholar
Schneider, R. (2014). Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press.Google Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin, Heidelberg.10.1007/978-3-540-78859-1CrossRefGoogle Scholar
Schönhöfer, P. (2014). Fractal geometries: scaling of intrinsic volumes. Masters Thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg.Google Scholar
Schönhöfer, P. and Mecke, K. (2015). The shape of anisotropic fractals: scaling of Minkowski functionals. In Fractal Geometry and Stochastics V, Birkhäuser/Springer, Cham, pp. 39–52.10.1007/978-3-319-18660-3_3CrossRefGoogle Scholar
Van den Berg, J. and Ermakov, A. (1996). A new lower bound for the critical probability of site percolation on the square lattice. Random Structures Algorithms 8, 199–212.3.0.CO;2-T>CrossRefGoogle Scholar
White, D. G. (2001). On the value of the critical point in fractal percolation. Random Structures Algorithms 18, 332–345.CrossRefGoogle Scholar
Winter, S. (2008). Curvature measures and fractals. Diss. Math. 453, 1–66.Google Scholar
Winter, S. and Zähle, M. (2013). Fractal curvature measures of self-similar sets. Adv. Geom. 13, 229–244.10.1515/advgeom-2012-0026CrossRefGoogle Scholar
Zähle, M. (2011). Lipschitz–Killing curvatures of self-similar random fractals. Trans. Amer. Math. Soc. 363, 2663–2684.10.1090/S0002-9947-2010-05198-0CrossRefGoogle Scholar