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**Harvard**

Broman, E. och Camia, F. (2010) *Universal Behavior of Connectivity Properties in Fractal Percolation Models*.

** BibTeX **

@article{

Broman2010,

author={Broman, Erik and Camia, F.},

title={Universal Behavior of Connectivity Properties in Fractal Percolation Models},

journal={Electronic Journal of Probability},

issn={1083-6489},

volume={15},

pages={1394-1414},

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 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 >= 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.},

year={2010},

keywords={random fractals, fractal percolation, continuum percolation, Mandelbrot, percolation, phase transition, crossing probability, discontinuity, Brownian loop soup, Poisson Boolean Model, random cantor sets, dimensions },

}

** RefWorks **

RT Journal Article

SR Electronic

ID 127611

A1 Broman, Erik

A1 Camia, F.

T1 Universal Behavior of Connectivity Properties in Fractal Percolation Models

YR 2010

JF Electronic Journal of Probability

SN 1083-6489

VO 15

SP 1394

OP 1414

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 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 >= 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.

LA eng

DO 10.1214/EJP.v15-805

LK http://dx.doi.org/10.1214/EJP.v15-805

LK http://publications.lib.chalmers.se/records/fulltext/local_127611.pdf

OL 30