### 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) |
---|---|

Pages (from-to) | 1394-1414 |

Number of pages | 21 |

Journal | Electronic Journal of Probability |

Volume | 15 |

DOIs | |

State | Published - Jan 1 2010 |

### Fingerprint

### 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

### Cite this

*Electronic Journal of Probability*,

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

**Universal behavior of connectivity properties in fractal percolation models.** / Broman, Erik I.; Camia, Federico.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 15, pp. 1394-1414. https://doi.org/10.1214/EJP.v15-805

}

TY - JOUR

T1 - Universal behavior of connectivity properties in fractal percolation models

AU - Broman, Erik I.

AU - Camia, Federico

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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 .

AB - 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 .

KW - Brownian loop soup

KW - Continuum percolation

KW - Crossing probability

KW - Discontinuity

KW - Fractal percolation

KW - Mandelbrot percolation

KW - Phase transition

KW - Poisson Boolean Model

KW - Random fractals

UR - http://www.scopus.com/inward/record.url?scp=78649706943&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78649706943&partnerID=8YFLogxK

U2 - 10.1214/EJP.v15-805

DO - 10.1214/EJP.v15-805

M3 - Article

AN - SCOPUS:78649706943

VL - 15

SP - 1394

EP - 1414

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -