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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: [email protected]
**Postal address: Karlsruhe Institute of Technology, Department of Mathematics, 76128 Karlsruhe, Germany. Email: [email protected]
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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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