 |
|
|
| | | | | |
|
|
|
|
|
The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space
|
Johan Harald Tykesson, Chalmers University of Technology
|
Abstract
We consider the Poisson Boolean model of continuum percolation with
balls of fixed radius R in n-dimensional hyperbolic space
Hn. Let λ be the intensity of the underlying Poisson
process, and let NC denote the number of unbounded components in
the covered region. For the model in any dimension we show that
there are intensities such that NC=∞ a.s. if R is big
enough. In H2 we show a stronger result: for any R there are
two intensities λc and λu where
0<λc<λu<∞, such that NC=0 for λ in
[0,λc], NC=∞ for λ in (λc,λu)
and NC=1 for λ in [λu,∞).
|
Full text: PDF
Pages: 1379-1401
Published on: November 4, 2007
|
Bibliography
- P. Albin. Private communication.
- K.S. Alexander. The RSW theorem for continuum percolation and the
CLT for Euclidean minimal spanning trees.
Ann. Appl. Probab. 6 (1996), no. 2, 466-494.
Math. Review 97f:60204
- K.B. Athreya and P.E. Ney. Branching processes. Springer
Verlag, 1972.
- I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Critical
percolation on any nonamenable group has no infinite
clusters. Ann. Probab. 27 (1999), 1347-1356.
Math. Review 2000k:60197
- I. Benjamini, R. Lyons, Y. Peres, and O. Schramm. Group-invariant
percolation on graphs. Geom. Funct. Anal. 9 (1999),
29-66.
Math. Review 99m:60149
- I. Benjamini and O. Schramm. Percolation in the hyperbolic
plane. J. Amer. Math. Soc. 24 (2001), 487-507.
Math. Review 2002h:82049
- I. Benjamini and O. Schramm. Percolation beyond ${mathbb Z}^d$,
many questions and a few answers. Electronic Commun. Probab.
1 (1996), 71-82.
Math. Review 97j:60179
- R.M. Burton and M. Keane. Density and uniqueness in
percolation. Comm. Math. Phys.
121 (1989), 501-505.
Math. Review 90g:60090
- J.W. Cannon, W.J. Floyd, R. Kenyon and W.R. Parry. Hyperbolic
geometry. In Flavors of geometry, pp. 59-115. Cambridge
University Press, 1997.
- A. Erdélyi, W. Magnus, F. Oberhettinger, and
F.G. Tricomi. Higher Transcendental Functions, Vol
I. McGraw-Hill, 1953.
- A. Erdélyi, W. Magnus, F. Oberhettinger, and
F.G. Tricomi. Higher Transcendental Functions, Vol
II. McGraw-Hill, 1953.
- G. Grimmett, Percolation (2nd ed.). Springer-Verlag, 1999.
- O. Häggström. Infinite clusters in dependent automorphism
invariant percolation on trees. Ann. Probab.
25 (1997), 1423-1436.
Math. Review 98f:60207
- O. Häggström and J. Jonasson. Uniqueness and non-uniqueness in
percolation theory. Probability Surveys
3 (2006), 289-344.
Math. Review 2007m:60297
- P. Hall. On continuum percolation. Ann. Probab.
13 (1985), 1250-1266.
Math. Review 87f:60018
- J. Jonasson. Hard-sphere percolation: Some positive answers in the
hyperbolic plane and on the integer lattice. Preprint 2001.
- I. Pak and T. Smirnova-Nagnibeda. On non-uniqueness of percolation
on nonamenable Cayley graphs. C. R. Acad. Sci. Paris Sr. I
Math.
330 (2000), 495-500.
Math. Review 2000m:60116
- R. Meester and R. Roy. Continuum Percolation. Cambridge
University Press, 1996.
- A.P. Prudnikov, Y.A. Brychkov, and O.I. Marichev. Integrals
and Series. Volume 1: Elementary Functions. Gordon and Breach
Science Publishers, 1986.
- J.G. Ratcliffe. Foundations of hyperbolic
manifolds. Springer-Verlag, 2006.
- A. Sarkar. Co-existence of the occupied and vacant phase in
Boolean models in three or more dimensions. Adv. Appl. Prob.
29 (1997), 878-889.
Math. Review 99a:60007
- R.H. Schonmann. Stability of infinite clusters in supercritical
percolation. Probab. Th. Rel. Fields
113 (1999), 287-300.
Math. Review 99k:60252
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context