Universal behavior of connectivity properties in fractal percolation models

Erik I. Broman, Federico Camia

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1394-1414
Number of pages21
JournalElectronic Journal of Probability
Volume15
DOIs
StatePublished - Jan 1 2010

Fingerprint

Connected Components
Fractal
Connectivity
Random Fractals
Fractal Set
Random Sets
Stochastic Geometry
Boolean Model
Scale Invariance
Poisson Model
Scale Invariant
Model
Critical point
Continuum
Phase Transition
Imply

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

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

In: Electronic Journal of Probability, Vol. 15, 01.01.2010, p. 1394-1414.

Research output: Contribution to journalArticle

@article{19ad930cefbf4890b7f499f19482f28a,
title = "Universal behavior of connectivity properties in fractal percolation models",
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 .",
keywords = "Brownian loop soup, Continuum percolation, Crossing probability, Discontinuity, Fractal percolation, Mandelbrot percolation, Phase transition, Poisson Boolean Model, Random fractals",
author = "Broman, {Erik I.} and Federico Camia",
year = "2010",
month = "1",
day = "1",
doi = "10.1214/EJP.v15-805",
language = "English (US)",
volume = "15",
pages = "1394--1414",
journal = "Electronic Journal of Probability",
issn = "1083-6489",
publisher = "Institute of Mathematical Statistics",

}

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 -