Geometric functionals of fractal percolation

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 beyond the fractal dimension not so much is known about the geometry of these sets. 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).