## 1 The number of unbounded components in the Poisson-Boolean model in H 2 (2008)

### BibTeX

@MISC{Tykesson081the,

author = {Johan Tykesson},

title = {1 The number of unbounded components in the Poisson-Boolean model in H 2},

year = {2008}

}

### OpenURL

### Abstract

We consider the Poisson-Boolean model with unit radius in the hyperbolic disc H 2. Let λ be the intensity of the underlying Poisson process, and let NC denote the number of unbounded components of the covered region. We show that there are two intensities λc and λu, 0 < λc < λu < ∞, such that NC = 0 for λ ∈ (0,λc], NC = ∞ for λ ∈ (λc,λu), and NC = 1 for λ ∈ [λu, ∞). Corresponding results, due to Benjamini, Lyons, Peres and Schramm, are available for Bernoulli bond and site percolation on certain nonamenable transitive graphs, and we use many of their techniques in our proofs. 1

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Citation Context ...teger k, let { 0 if P[Dn|X(S(z, k))] ≤ 1/2 1n,z,k := 1 if P[Dn|X(S(z, k))] > 1/2 Thus 1n,z,k is the best guess of 1Dn given the configuration of the Poisson process in S(z, k). By Lévy’s 0-1-law (see =-=[9]-=-, page 263) we get for fixed z that lim k→∞ 1n,z,k = 1Dn a.s. (4.8) Let z1, z2, ... be a sequence of points such that for each k, S(z, k) and S(zk, k) do not intersect. Since (1Dn,1n,zk,k) has the sam... |

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Citation Context ...n We begin by describing the unit radius version of the so called Poisson-Boolean model in R 2 , arguably the most studied continuum percolation model. For a detailed study of this model, we refer to =-=[15]-=-. Let X be a Poisson point process in R 2 with some intensity λ. At each point of X, place a closed ball with unit radius. Let C be the union of all balls, and V be the complement of C. The sets V and... |

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Citation Context ... that the graph is nonamenable. Benjamini and Schramm [6] have made the following general conjecture: Conjecture 2.5 If G is transitive, then pu > pc if and only if G is nonamenable. Burton and Keane =-=[7]-=- solved one part of Conjecture 2.5: Theorem 2.6 Assume that G is transitive and amenable. If ω is a p-Bernoulli percolation on G with p > pc, then ω contains a unique infinite cluster a.s. In fact, th... |

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Citation Context ... . Definition 2.4 A bounded degree graph G = (V, E) is said to be amenable if κV (G) = 0 (or equivalently κE(G) = 0). If instead κV (G) > 0 we say that the graph is nonamenable. Benjamini and Schramm =-=[6]-=- have made the following general conjecture: Conjecture 2.5 If G is transitive, then pu > pc if and only if G is nonamenable. Burton and Keane [7] solved one part of Conjecture 2.5: Theorem 2.6 Assume... |

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Citation Context ...nt bond percolation on G, then for any u ∈ V ∑ E[m(u, v, ω)] = ∑ E[m(v, u, ω)]. v∈V v∈V Theorem 2.11 and the proof we present below is due to Benjamini, Lyons, Peres and Schramm [3]. The same authors =-=[4]-=- prove the mass transport principle for a wider class of transitive graphs. The first version of the mass transport principle was proved by Häggström [11], for homogeneous trees. In words, the mass tr... |

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Citation Context ...Lyons, Peres and Schramm [3]. The same authors [4] prove the mass transport principle for a wider class of transitive graphs. The first version of the mass transport principle was proved by Häggström =-=[11]-=-, for homogeneous trees. In words, the mass transport principle says that the expected amount of mass transported out of the vertex v is the same as the expected amount of mass transported into it. Pr... |

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Citation Context ...n automorphism invariant bond percolation on G, then for any u ∈ V ∑ E[m(u, v, ω)] = ∑ E[m(v, u, ω)]. v∈V v∈V Theorem 2.11 and the proof we present below is due to Benjamini, Lyons, Peres and Schramm =-=[3]-=-. The same authors [4] prove the mass transport principle for a wider class of transitive graphs. The first version of the mass transport principle was proved by Häggström [11], for homogeneous trees.... |

31 | Percolation in the hyperbolic plane
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Citation Context ...l. Thus t = 0, a contradiction. ✷ The other direction of Conjecture 2.5 has only been partially solved. Here is one such result that will be of particular interest to us, due to Benjamini and Schramm =-=[5]-=-. This can be considered as the discrete analogue to our main theorem. First, another definition is needed. Definition 2.7 Let G = (V, E) be an infinite connected graph and for W ⊂ V let NW be the num... |

25 |
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Citation Context ... large enough, there are no infinite clusters in ω. But if there are no infinite clusters in ω, there are no unbounded components of V . Thus λ∗ c < ∞. To show λc > 0 we adapt an argument due to Hall =-=[13]-=-. Construct a branching process, whose members are points in H2 , as follows. The individual in the 0’th generation is taken to be the center of a ball with unit radius. Without loss of generality the... |

23 |
The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees
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Citation Context ...usters. One of the most basic facts in the theory of discrete percolation is the following theorem, a proof of which can be found in [12]. Theorem 2.1 There exists a critical probability pc = pc(G) ∈ =-=[0, 1]-=- such that { 0, p < pc Pp(C) = 1, p > pc A natural question to ask is: How many infinite clusters are there? The answer obviously depends on G and p. We will consider only transitive graphs. Definitio... |

22 |
Stability of infinite clusters in supercritical percolation
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(Show Context)
Citation Context ... to introduce another quantity of interest, alongside pc. 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 a.s. Schonmann =-=[16]-=- showed for all transitive graphs that for all p > pu, one has uniqueness. In view of Theorem 2.3 this means there are at most three phases for p ∈ [0, 1] regarding the number of infinite clusters, na... |

11 | Hyperbolic Geometry
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Citation Context ...) as r → 0 (3.3) µ(S(0, r)) = πr 2 + o(r 3 ) as r → 0. (3.4) Thus, at small scale, hyperbolic length and area are close to Euclidean length and area. For more elementary facts about H 2 , we refer to =-=[8]-=-.7 3.1 Mass transport Next, we present the mass transport principle for H 2 , due to Benjamini and Schramm [5]. It is essential for our results, thus we include a proof. First some preliminary defini... |

8 | Uniqueness and non-uniqueness in percolation theory
- Häggström, Jonasson
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(Show Context)
Citation Context ...be the event that p-Bernoulli bond percolation contains infinite clusters. One of the most basic facts in the theory of discrete percolation is the following theorem, a proof of which can be found in =-=[12]-=-. Theorem 2.1 There exists a critical probability pc = pc(G) ∈ [0, 1] such that { 0, p < pc Pp(C) = 1, p > pc A natural question to ask is: How many infinite clusters are there? The answer obviously d... |

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