Universal Behavior of Connectivity Properties in Fractal Percolation Models
Erik I Broman, Chalmers University of Technology
Federico Camia, Vrije Universiteit Amsterdam
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 ≥ 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 λ. 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 λ
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.
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