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 $d\geq2$. 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 $d\geq2$. 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.
Highlights
We introduce and study a natural class of random fractals that exhibit, in dimension d ≥ 2, such a connectivity phase transition: when a parameter increases continuously through a critical value, the connectivity suddenly breaks down and the random fractals become totally disconnected with probability one. (We remind the reader that a set is called totally disconnected if it contains no connected component larger than one point.) The fractals we study are defined as the complement of the union of sets generated by a Poisson point process of intensity λ times a scale invariant measure on a space of subsets of Rd
Examples of such random fractals include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry, the Brownian loop soup [18], and the models studied in [27]
The scale invariant (Poisson) Boolean model is a natural model for a porous medium with cavities on many different scales
Summary
Many deterministic constructions generating fractal sets have random analogues that produce random fractals which do not have the self-similarity of their non-random counterpart, but are statistically self-similar in the sense that enlargements of small parts have the same statistical distribution as the whole set. Our main result consists in showing that, when the intensity λ of the Poisson process is at its critical value, the random fractals are in the connected phase in the sense that they contain connected components larger than one point with probability one (see Theorem 2.4)
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