# Geometric functionals of fractal percolation

## TL;DR

This paper introduces geometric functionals for fractal percolation, providing explicit formulas and limits, which could help estimate the critical transition points in these complex fractal systems.

## Contribution

It develops a new framework of geometric functionals for fractal percolation, extending classical additive geometric measures to fractal models.

## Key findings

- Existence of limit geometric functionals for fractal percolation.
- Explicit formulas for the geometric functionals and their finite approximations.
- Potential application in estimating percolation thresholds.

## 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 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.