Dimension (in)equalities and H\"older continuous curves in fractal percolation

TL;DR

This paper explores the relationships between fractal dimensions of the limiting set in fractal percolation, showing that in two dimensions, the large connected components form H"older continuous curves and all curves have Hausdorff dimension greater than one.

Contribution

It establishes new links between fractal dimensions and geometric structures in fractal percolation, including the characterization of large components as H"older curves and dimension bounds for all curves.

Findings

Large connected components are unions of H"older continuous curves with the same exponent.

The Hausdorff dimension of any curve in the set exceeds 1.

Relationships between various fractal dimensions of the limiting set and its components.

Abstract

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial H\"older continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.