Random soups, carpets and fractal dimensions

TL;DR

This paper investigates the Hausdorff dimensions of certain random planar fractal sets generated by Poissonian processes, establishing their deterministic nature and analyzing their behavior at low intensities, with applications to conformal loop ensembles.

Contribution

It proves that the Hausdorff dimensions of these fractals are deterministic and equal to their expectation, and provides estimates for their low-intensity limits, extending understanding of Poissonian fractal models.

Findings

Hausdorff dimensions are deterministic and match expectation dimensions

Dimensions are estimated in the low-intensity limit

Results apply to conformal loop ensembles with known expectation dimensions

Abstract

We study some properties of a class of random connected planar fractal sets induced by a Poissonian scale-invariant and translation-invariant point process. Using the second-moment method, we show that their Hausdorff dimensions are deterministic and equal to their expectation dimension. We also estimate their low-intensity limiting behavior. This applies in particular to the "conformal loop ensembles" defined via Poissonian clouds of Brownian loops for which the expectation dimension has been computed by Schramm, Sheffield and Wilson.