Universal Behavior of Connectivity Properties in Fractal Percolation Models

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 ≥ 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 ̧. 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 ̧ 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 ≥ 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. AMS subject classification: 60D05, 28A80, 60K35 ∗Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden. E-mail: [email protected] †Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. E-mail: fede@ few.vu.nl