Universal Behavior of Connectivity Properties in Fractal Percolation Models
Federico Camia (Vrije Universiteit Amsterdam)
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.
Full Text: Download PDF | View PDF online (requires PDF plugin)
Pages: 1394-1414
Publication Date: September 19, 2010
DOI: 10.1214/EJP.v15-805
References
- M. Aizenmann and G. Grimmett, Strict Monotonicity for Critical Points in Percolation and Ferromagnetic Models, J. Stat. Phys. 63 (1991) 817-835 Math. Review 92i:82060
- H. Bierme and A. Estrade, Covering the whole space with Poisson random ball in preparation (2010)
- E.I. Broman and F. Camia, Large-$N$ limit of crossing probabilities, discontinuity, and asymptotic behavior of threshold values in Mandelbrot's fractal percolation process, Electron. J. Probab. 13 (2008) 980-999 Math. Review 2009g:60130
- R.M. Burton and M. Keane, Density and Uniqueness in Percolation, Comm. Math. Phys. 121 (1989) 501-505 Math. Review 90g:60090
- J.T. Chayes and L. Chayes, The large-$N$ limit of the threshold values in Mandelbrot's fractal percolation process, J.Phys.A: Math. Gen. 22 (1989) L501--L506 Math. Review 90h:82044
- J.T. Chayes, L. Chayes and R. Durrett, Connectivity Properties of Mandelbrot's Percolation Process, Probab. Theory Relat. Fields 77 (1988) 307-324 Math. Review 89d:60193
- J.T. Chayes, L. Chayes, E. Grannan and G. Swindle, Phase transitions in Mandelbrot's percolation process in three Probab. Theory Relat. Fields 90 (1991) 291-300 Math. Review 93a:60156
- L. Chayes, Aspects of the fractal percolation process, Progress in Probability 37 (1995) 113-143 Math. Review 97g:60131
- F.M. Dekking and G.R. Grimmett, Superbranching processes and projections of random Cantor sets, Probab. Theory Relat. Fields 78 (1988) 335-355 Math. Review 89f:60099
- F.M. Dekking and R.W.J. Meester, On the structure of Mandelbrot's percolation process and other Random Cantor sets J. Stat. Phys. 58 (1990) 1109-1126 Math. Review 91c:60140
- K.J. Falconer Fractal Geometry Second edition, Wiley, Chichester, 2003. Math. Review 2006b:28001
- K.J. Falconer and G.R. Grimmett, The critical point of fractal percolation in three and more dimensions, J. Phys. A: Math. Gen. 24 (1991) L491--L494 Math. Review 92g:82053
- K.J. Falconer and G.R. Grimmett, On the geometry of Random Cantor Sets and Fractal Percolation, J. Theor. Probab. 5 (1992) 465-485 Math. Review 94b:60115
- B. Freivogel and M. Kleban, A Conformal Field Theory for Eternal Inflation? J. High Energy Phys. 12 (2009) 019 Math. Review number not available
- G. Grimmett, Percolation Second edition, Springer-Verlag, Berlin 1999 Math. Review 2001a:60114
- S. Janson, Bounds of the distribution of extremal values of a scanning process, Stochastic Process. Appl. 18 (1984) 313-328. Math. Review 86f:60066
- G.F. Lawler, Conformally Invariant Processes in the Plane Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, 2005 Math. Review 2006i:60003
- G.F. Lawler and W. Werner, The Brownian loop soup, Probab. Theory Relat. Fields 128 (2004) 565-588. Math. Review 2005f:60176
- T.M. Liggett Stochastic Interacting Systems: Contact, Voter and Exclusion Processes Springer-Verlag, Berlin, 1999 Math. Review 2001g:60247
- T.M. Liggett, R.H. Schonmann and A.M. Stacey, Domination by product measures, Ann. Probab. 25 (1997) 71-95. Math. Review 98f:60095)
- B.B.~Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974) 331-358. Math. Review number not available.
- B.B. Mandelbrot, The Fractal Geometry of Nature W.H. Freeman, San Francisco (1983) Math. Review 84h:00021
- R.W.J. Meester, Connectivity in fractal percolation, 5 (1992) 775-789 Math. Review 93m:60201
- R. Meester and R. Roy, Continuum Percolation Cambridge University Press, New York, 1996. Math. Review 98d:60193
- M.V.Menshikov, S.Yu. Popov and M. Vachkovskaia, On the connectivity properties of the complementary set in fractal percolation models, Probab. Theory Relat. Fields 119 (2001) 176-186 Math. Review 2002d:60085
- M.V.Menshikov, S.Yu. Popov and M. Vachkovskaia, On a multiscale continuous percolation model with unbounded defects, Bull. Braz. Math. Soc. 34 (2003) 417-435. Math. Review 2005c:60132
- S.~Nacu and W.~Werner, Random soups, carpets and dimensions, J. London Math. Soc. to appear (2010) Math. Review number not available.
- M.E. Orzechowski, On the Phase Transition to Sheet Percolation in Random Cantor Sets J. Stat. Phys. 82 (1996) 1081-1098 Math. Review 97e:82022
- R. Schneider and W. Weil, Stochastic and Integral Geometry Springer-Verlag, Berlin, 2008. Math. Review 2010g:60002
- S. Sheffield and W. Werner, Conformal Loop Ensembles: Construction via Loop-soups, preprint arXiv:1006.2373v1
- S. Sheffield and W. Werner, Conformal Loop Ensembles: The Markovian Characterization, preprint arXiv:1006.2374v1
- D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications Second edition, Wiley, Chichester, 1985. Math. Review 88j:60034a
- W. Werner, SLEs as boundaries of clusters of Brownian loops, C. R. Math. Acad. Sci. Paris 337 (2003) 481-486 Math. Review 2005b:60221
- W. Werner, Some recent aspects of random conformally invariant systems, Les Houches Scool Proceedings: Session LXXXII, Mathematical Statistical Physics (2006) 57-98 Math. Review number not available.
- D.G. White, On fractal percolation in ${mathbb R}^2$, Statist. Probab. Lett. 45 (1999) 187-190 Math. Review 2000i:60117

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