Universal Behavior of Connectivity Properties in Fractal Percolation Models

TL;DR

This paper investigates the connectivity phase transition in various fractal percolation models, demonstrating that the emergence of large connected components at critical points is a universal, scale-invariance-driven phenomenon across dimensions.

Contribution

It introduces a class of scale-invariant continuum fractal percolation models and proves the universality of the connectivity transition behavior across different models and dimensions.

Findings

Connected components larger than one point appear at the critical parameter value.

The transition behavior is shown to be universal and independent of model specifics.

In 2D, the existence of large components implies a unique unbounded connected component.

Abstract

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of lambda that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d greater than or equal to 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for d=2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.