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Universal Behavior of Connectivity Properties in Fractal Percolation Models

Erik Broman (Institutionen för matematiska vetenskaper, matematik) ; F. Camia
Electronic Journal of Probability (1083-6489). Vol. 15 (2010), p. 1394-1414.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: random fractals, fractal percolation, continuum percolation, Mandelbrot, percolation, phase transition, crossing probability, discontinuity, Brownian loop soup, Poisson Boolean Model, random cantor sets, dimensions



Denna post skapades 2010-10-14. Senast ändrad 2015-07-02.
CPL Pubid: 127611

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Matematik

Chalmers infrastruktur