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Geometric functionals of fractal percolation

Vrijeme: 8.7.2020
Predavaonica: vir
Predavač: Steffen Winter, Karlsruhe Institute of Technology (KIT), Institute of Stochastics
Naziv: Geometric functionals of fractal percolation

The talk is scheduled in virtual setting, using Zoom platform. The talk will also be live-streamed via YouTube. During the meeting, the questions can be posed via chat or audio for participants in the meeting. Everybody interested is invited to participate in the Zoom meeting, with the limit of 100 participants, or to follow the live-broadcast. The link to Zoom meeting will be available at least one day before the talk, and the link to live broadcast 10-15 minutes before the talk, on this page.

The talk will be in English.

Fractal percolation is a family of random self-similar sets suggested by Mandelbrot in the seventies to model certain aspects of turbulence. It exhibits a dramatic topological phase transition, changing abruptly from a dust-like structure to the appearance of a system spanning cluster. The transition points are unknown and difficult to estimate, and the geometry of these sets is not completely understood. It is a natural question whether geometric functionals such as intrinsic volumes can provide further insights.
We study some geometric functionals of the fractal percolation process F, which arise as suitably rescaled limits of intrinsic volumes of finite approximations of F. We establish the almost sure existence of these limit functionals, clarify their structure and obtain explicit formulas for their expectations and variances as well as for their finite approximations. The approach is similar to fractal curvatures but in contrast the new functionals can be determined explicitly and approximated well from simulations.
Joint work with Michael Klatt (Princeton).

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