### 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 .

Original language | English (US) |
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Pages (from-to) | 1394-1414 |

Number of pages | 21 |

Journal | Electronic Journal of Probability |

Volume | 15 |

DOIs | |

State | Published - Jan 1 2010 |

### Keywords

- Brownian loop soup
- Continuum percolation
- Crossing probability
- Discontinuity
- Fractal percolation
- Mandelbrot percolation
- Phase transition
- Poisson Boolean Model
- Random fractals

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Electronic Journal of Probability*,

*15*, 1394-1414. https://doi.org/10.1214/EJP.v15-805