Fractal percolation exhibits a dramatic topological phase transition, changing abruptly
from a dust-like set to a system-spanning cluster. Where this transition occurs is unknown
and notoriously difficult to estimate. In continuum percolation, the percolation thresholds have been approximated well using Minkowski functionals. Motivated by the question of whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process.
M. A. Klatt and S. Winter. Geometric functionals of fractal percolation. Adv. Appl. Probab. 52:1085–1126 (2020)
We establish the existence of the limit functionals and obtain explicit formulas for them as well as for their finite approximations (in excellent agreement with simulation result). They provide new geometric insights into fractal percolation. For example, we provide a mathematically rigorous notion of fractal subdimensions that have been previously defined in physics as an approach to refined information about fractal sets beyond fractal dimension.
My code to simulate fractal percolation and to analyze our geometric functionals is freely available at GitHub.