## The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space (2007)

Citations: | 3 - 2 self |

### BibTeX

@MISC{Tykesson07thenumber,

author = {Johan Tykesson},

title = {The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space},

year = {2007}

}

### OpenURL

### Abstract

### Citations

903 | Probability: Theory and Examples - Durrett - 1996 |

236 |
Continuum Percolation
- Meester, Roy
- 1996
(Show Context)
Citation Context ... We begin by describing the fixed radius version of the so called Poisson Boolean model in R n , arguably the most studied continuum percolation model. For a detailed study of this model, we refer to =-=[19]-=-. Let X be a Poisson point process in R n with some intensity λ. At each point of X, place a closed ball of radius R. Let C be the union of all balls, and V be the complement of C. The sets V and C wi... |

148 |
Foundations of Hyperbolic Manifolds
- Ratcliffe
- 1994
(Show Context)
Citation Context ...use of the fact that for any ǫ ∈ (0, r) there is a constant K(ǫ, n) > 0 independent of r such that (3.3) µ(S(0, r) \ S(0, r − ǫ)) ≥ K(ǫ, n)µ(S(0, r)) for all r. For more facts about H n , we refer to =-=[21]-=-.CONTINUUM PERCOLATION IN HYPERBOLIC SPACE 5 3.1. Mass transport. Next, we present an essential ingredient to our proofs in H 2 , the mass transport principle which is due to Benjamini and Schramm [6... |

120 |
Density and uniqueness in percolation
- Burton, Keane
- 1989
(Show Context)
Citation Context ...ollowing general conjecture: Conjecture 2.1. If G is transitive, then pu > pc if and only if G is nonamenable. Of course, one direction of the conjecture is the well-known theorem by Burton and Keane =-=[8]-=- which says that any transitive, amenable graph G has a unique infinite cluster for all p > pc. The other direction of Conjecture 2.1 has only been partially solved. Here is one such result that will ... |

83 | Percolation beyond , many questions and a few answers
- dBENJAMINI, SCHRAMM
- 1996
(Show Context)
Citation Context ...efined as κV (G) := infW |∂V W | |W | where the infimum ranges over all finite connected subsets W of V . A bounded degree graph G = (V, E) is said to be amenable if κV (G) = 0. Benjamini and Schramm =-=[7]-=- have made the following general conjecture: Conjecture 2.1. If G is transitive, then pu > pc if and only if G is nonamenable. Of course, one direction of the conjecture is the well-known theorem by B... |

82 | Groupinvariant percolation on graphs - Benjamini, Lyons, et al. - 1999 |

53 | In clusters in dependent automorphism invariant percolation on trees - Haggstrom - 1997 |

41 | Critical percolation on any nonamenable group has no infinite clusters
- Benjamini, Lyons, et al.
- 1998
(Show Context)
Citation Context ...rs in Bernoulli bond percolation. For H 2 we also show that the model does not percolate on λc. The discrete analogue of this theorem is due to Benjamini, Lyons, Peres and Schramm and can be found in =-=[4]-=-. It turns out that several techniques from the aforementioned papers are possible to adopt to the continuous setting in H 2 . There is also a discrete analogue to the main result in H n . In [18], Pa... |

34 | Percolation in the hyperbolic plane
- Benjamini, Schramm
(Show Context)
Citation Context ...inear isoperimetric inequality says that the circumference of a bounded set is always bigger than the area of the set. The main result in H 2 is inspired by a theorem due to Benjamini and Schramm. In =-=[6]-=- they show that for a large class of nonamenable planar transitive graphs, there are infinitely many infinite clusters for some parameters in Bernoulli bond percolation. For H 2 we also show that the ... |

28 | On continuum percolation - Hall - 1985 |

25 |
Stability of infinite clusters in supercritical percolation
- Schonmann
- 1999
(Show Context)
Citation Context ...t there is 1 or ∞ infinite clusters for transitive graphs. If we let pu = pu(G) be the infimum of the set of p ∈ [0, 1] such that p-Bernoulli bond percolation has a unique infinite cluster, Schonmann =-=[22]-=- showed for all transitive graphs, one has uniqueness for all p > pu. Thus there are at most three phases for p ∈ [0, 1] regarding the number of infinite clusters, namely one for which this number is ... |

24 | The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees - Alexander - 1996 |

24 | Percolation beyond Zd , Many Questions and A Few Answers - Benjamini, Lyons, et al. - 1996 |

23 | On non-uniqueness of percolation on nonamenable Cayley graphs
- Pak, Smirnova-Nagnibeda
(Show Context)
Citation Context ...d in [4]. It turns out that several techniques from the aforementioned papers are possible to adopt to the continuous setting in H 2 . There is also a discrete analogue to the main result in H n . In =-=[18]-=-, Pak and Smirnova show that for certain Cayley graphs, there is a non-uniqueness phase for the number of unbounded components. In this case, while it is still possible to adopt their main idea to the... |

15 | Hyperbolic geometry - Cannon, Floyd, et al. - 1997 |

8 | Uniqueness and non-uniqueness in percolation theory
- Häggström, Jonasson
- 2006
(Show Context)
Citation Context ... follows. In section 2 we give a very short review of uniqueness and non-uniqueness results for infinite clusters in Bernoulli percolation on graphs (for a more extensive review, see the survey paper =-=[15]-=-), including the results by Benjamini, Lyons, Peres, Schramm, Pak and Smirnova. In section 3 we review some elementary properties of H n . In section 4 we introduce the model, and give some basic resu... |

7 | Uniqueness and non-uniqueness in percolation theory, Probab - Häggström, Jonasson |

6 | Percolation (2nd ed - Grimmett - 1999 |

5 | Hard-sphere percolation: Some positive answers in the hyperbolic plane and on the integer lattice - Jonasson - 2001 |

3 |
private communication
- Albin
(Show Context)
Citation Context ...Let I be the event that p-Bernoulli bond percolation contains infinite clusters. One of the most basic facts in the theory of discrete percolation is that there is a critical probability pc = pc(G) ∈ =-=[0, 1]-=- such that Pp(I) = 0 for p < pc(G) andCONTINUUM PERCOLATION IN HYPERBOLIC SPACE 3 Pp(I) = 1 for p > pc(G). What happens on pc depends on the graph. Above pc it is known that there is 1 or ∞ infinite ... |

2 | Integrals and series, Volume 1: elementary functions - Prudnikov, Brychkov, et al. - 1992 |

2 | Co-existence of the occupied and vacant phase in Boolean models in three or more dimensions - Sarkar - 1997 |