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.
Electron. J. Probab.
15:
1394-1414
(2010).
DOI: 10.1214/EJP.v15-805
H. Bierme and A. Estrade, Covering the whole space with Poisson random ball. In preparation (2010) 1277.60094H. Bierme and A. Estrade, Covering the whole space with Poisson random ball. In preparation (2010) 1277.60094
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 MR2413292 1191.60109 10.1214/EJP.v13-511 euclid.ejp/1464819106E.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 MR2413292 1191.60109 10.1214/EJP.v13-511 euclid.ejp/1464819106
R.M. Burton and M. Keane, Density and Uniqueness in Percolation, Comm. Math. Phys. 121 (1989) 501-505 0662.60113 10.1007/BF01217735 euclid.cmp/1104178143R.M. Burton and M. Keane, Density and Uniqueness in Percolation, Comm. Math. Phys. 121 (1989) 501-505 0662.60113 10.1007/BF01217735 euclid.cmp/1104178143
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 0849.60092 10.1016/0304-4149(95)00071-2J.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 0849.60092 10.1016/0304-4149(95)00071-2
J.T. Chayes, L. Chayes and R. Durrett, Connectivity Properties of Mandelbrot's Percolation Process, Probab. Theory Relat. Fields 77 (1988) 307-324 0621.60110 10.1007/BF00319291J.T. Chayes, L. Chayes and R. Durrett, Connectivity Properties of Mandelbrot's Percolation Process, Probab. Theory Relat. Fields 77 (1988) 307-324 0621.60110 10.1007/BF00319291
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 0734.60100 10.1007/BF01193747J.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 0734.60100 10.1007/BF01193747
L. Chayes, Aspects of the fractal percolation process, Progress in Probability 37 (1995) 113-143 0844.60091L. Chayes, Aspects of the fractal percolation process, Progress in Probability 37 (1995) 113-143 0844.60091
F.M. Dekking and G.R. Grimmett, Superbranching processes and projections of random Cantor sets, Probab. Theory Relat. Fields 78 (1988) 335-355 0628.60091 10.1007/BF00334199F.M. Dekking and G.R. Grimmett, Superbranching processes and projections of random Cantor sets, Probab. Theory Relat. Fields 78 (1988) 335-355 0628.60091 10.1007/BF00334199
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 0714.60102 10.1007/BF01026566F.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 0714.60102 10.1007/BF01026566
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 MR1117858 0728.60102 10.1088/0305-4470/24/9/007K.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 MR1117858 0728.60102 10.1088/0305-4470/24/9/007
K.J. Falconer and G.R. Grimmett, On the geometry of Random Cantor Sets and Fractal Percolation, J. Theor. Probab. 5 (1992) 465-485 0752.60007 10.1007/BF01060430K.J. Falconer and G.R. Grimmett, On the geometry of Random Cantor Sets and Fractal Percolation, J. Theor. Probab. 5 (1992) 465-485 0752.60007 10.1007/BF01060430
G. Grimmett, Percolation Second edition, Springer-Verlag, Berlin 1999 MR1707339 0926.60004G. Grimmett, Percolation Second edition, Springer-Verlag, Berlin 1999 MR1707339 0926.60004
S. Janson, Bounds of the distribution of extremal values of a scanning process, Stochastic Process. Appl. 18 (1984) 313-328. 0549.60066 10.1016/0304-4149(84)90303-XS. Janson, Bounds of the distribution of extremal values of a scanning process, Stochastic Process. Appl. 18 (1984) 313-328. 0549.60066 10.1016/0304-4149(84)90303-X
G.F. Lawler and W. Werner, The Brownian loop soup, Probab. Theory Relat. Fields 128 (2004) 565-588. 1049.60072 10.1007/s00440-003-0319-6G.F. Lawler and W. Werner, The Brownian loop soup, Probab. Theory Relat. Fields 128 (2004) 565-588. 1049.60072 10.1007/s00440-003-0319-6
T.M. Liggett, R.H. Schonmann and A.M. Stacey, Domination by product measures, Ann. Probab. 25 (1997) 71-95. 0882.60046 10.1214/aop/1024404279 euclid.aop/1024404279T.M. Liggett, R.H. Schonmann and A.M. Stacey, Domination by product measures, Ann. Probab. 25 (1997) 71-95. 0882.60046 10.1214/aop/1024404279 euclid.aop/1024404279
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. 0289.76031 10.1017/S0022112074000711B.B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974) 331-358. 0289.76031 10.1017/S0022112074000711
R.W.J. Meester, Connectivity in fractal percolation, 5 (1992) 775-789 MR1182680 0782.60079 10.1007/BF01058729R.W.J. Meester, Connectivity in fractal percolation, 5 (1992) 775-789 MR1182680 0782.60079 10.1007/BF01058729
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 1001.60104 10.1007/PL00008757M.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 1001.60104 10.1007/PL00008757
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. MR2045167 1056.60099 10.1007/s00574-003-0022-3M.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. MR2045167 1056.60099 10.1007/s00574-003-0022-3
S. Nacu and W. Werner, Random soups, carpets and dimensions, J. London Math. Soc. to appear (2010) 1223.28012 10.1112/jlms/jdq094S. Nacu and W. Werner, Random soups, carpets and dimensions, J. London Math. Soc. to appear (2010) 1223.28012 10.1112/jlms/jdq094
M.E. Orzechowski, On the Phase Transition to Sheet Percolation in Random Cantor Sets J. Stat. Phys. 82 (1996) 1081-1098 1042.82561 10.1007/BF02179803M.E. Orzechowski, On the Phase Transition to Sheet Percolation in Random Cantor Sets J. Stat. Phys. 82 (1996) 1081-1098 1042.82561 10.1007/BF02179803
R. Schneider and W. Weil, Stochastic and Integral Geometry Springer-Verlag, Berlin, 2008. 1175.60003R. Schneider and W. Weil, Stochastic and Integral Geometry Springer-Verlag, Berlin, 2008. 1175.60003
S. Sheffield and W. Werner, Conformal Loop Ensembles: Construction via Loop-soups, preprint arXiv:1006.2373v1 1006.2373v1S. Sheffield and W. Werner, Conformal Loop Ensembles: Construction via Loop-soups, preprint arXiv:1006.2373v1 1006.2373v1
S. Sheffield and W. Werner, Conformal Loop Ensembles: The Markovian Characterization, preprint arXiv:1006.2374v1 1006.2374v1S. Sheffield and W. Werner, Conformal Loop Ensembles: The Markovian Characterization, preprint arXiv:1006.2374v1 1006.2374v1
D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications Second edition, Wiley, Chichester, 1985. 1155.60001D. Stoyan, W.S. Kendall and J. Mecke, Stochastic geometry and its applications Second edition, Wiley, Chichester, 1985. 1155.60001
W. Werner, SLEs as boundaries of clusters of Brownian loops, C. R. Math. Acad. Sci. Paris 337 (2003) 481-486 1029.60085 10.1016/j.crma.2003.08.003W. Werner, SLEs as boundaries of clusters of Brownian loops, C. R. Math. Acad. Sci. Paris 337 (2003) 481-486 1029.60085 10.1016/j.crma.2003.08.003
W. Werner, Some recent aspects of random conformally invariant systems, Les Houches Scool Proceedings: Session LXXXII, Mathematical Statistical Physics (2006) 57-98 1370.60142W. Werner, Some recent aspects of random conformally invariant systems, Les Houches Scool Proceedings: Session LXXXII, Mathematical Statistical Physics (2006) 57-98 1370.60142
D.G. White, On fractal percolation in ${mathbb R}^2$, Statist. Probab. Lett. 45 (1999) 187-190 0960.60098 10.1016/S0167-7152(99)00058-9D.G. White, On fractal percolation in ${mathbb R}^2$, Statist. Probab. Lett. 45 (1999) 187-190 0960.60098 10.1016/S0167-7152(99)00058-9