The existence phase transition for scale invariant Poisson random fractal models

TL;DR

This paper investigates the phase transition in scale invariant random fractal models, precisely identifying the critical point and showing that many models are empty at this threshold, including the Poisson Boolean model and Brownian loop soup.

Contribution

It determines the exact critical point for a broad class of scale invariant fractal models and proves emptiness at criticality without restricting to open set models.

Findings

Exact critical point identified for all models under weak assumptions

Models in the subclass are empty at the critical point

Includes scale invariant Poisson Boolean model and Brownian loop soup

Abstract

In this paper we study the existence phase transition of scale invariant random fractal models. We determine the exact value of the critical point of this phase transition for all models satisfying some weak assumptions. In addition, we show that for a large subclass, the fractal model is in the empty phase at the critical point. This subclass of models includes the scale invariant Poisson Boolean model and the Brownian loop soup. In contrast to earlier results in the literature, we do not need to restrict our attention to random fractal models generated by open sets.