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2010 Universal Behavior of Connectivity Properties in Fractal Percolation Models
Erik Broman, Federico Camia
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Electron. J. Probab. 15: 1394-1414 (2010). DOI: 10.1214/EJP.v15-805

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.

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Erik Broman. Federico Camia. "Universal Behavior of Connectivity Properties in Fractal Percolation Models." Electron. J. Probab. 15 1394 - 1414, 2010. https://doi.org/10.1214/EJP.v15-805

Information

Accepted: 19 September 2010; Published: 2010
First available in Project Euclid: 1 June 2016

zbMATH: 1225.60023
MathSciNet: MR2721051
Digital Object Identifier: 10.1214/EJP.v15-805

Subjects:
Primary: 60D05
Secondary: 28A80, 60K35

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