Recent Progress on Fractal Percolation

TL;DR

This survey reviews recent advances in fractal percolation, focusing on dimension theory and connectivity properties, highlighting new findings about the Assouad spectrum and intermediate dimensions of the limiting set.

Contribution

It provides a comprehensive overview of recent research in fractal percolation, emphasizing the dimension theory and geometric properties of the limiting set and its components.

Findings

Assouad spectrum and intermediate dimensions are constant and equal to the box dimension for non-trivial components

Classical results on connectivity properties are included

The survey highlights recent progress in understanding the geometry of fractal percolation sets

Abstract

This is a survey paper about the fractal percolation process, also known as Mandelbrot percolation. It is intended to give a general breadth overview of more recent research in the topic, but also includes some of the more classical results, for example related to the connectivity properties. Particular emphasis is put on the dimension theory of the limiting set and also on the geometry of the non-trivial connected components in the supercritical regime. In particular, we show that both the Assouad spectrum and intermediate dimensions of the non-trivial connected components are constant equal to its box dimension despite its Hausdorff, box and Assouad dimensions known to being distinct.