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Universal Behavior of Connectivity Properties in Fractal Percolation Models

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Universal Behavior of Connectivity Properties in Fractal Percolation Models
2.	Creator	Author's name, affiliation, country	Erik I Broman; Chalmers University of Technology; Sweden
2.	Creator	Author's name, affiliation, country	Federico Camia; Vrije Universiteit Amsterdam; Netherlands
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	random fractals, fractal percolation, continuum percolation, Mandelbrot percolation, phase transition, crossing probability, discontinuity, Brownian loop soup, Poisson Boolean Model
3.	Subject	Subject classification	60D05, 28A80, 60K35
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2010-09-19
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/805
10.	Identifier	Digital Object Identifier	10.1214/EJP.v15-805
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 15
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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