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**1 - 9**of**9**### CONTINUUM PERCOLATION AT AND ABOVE THE UNIQUENESS TRESHOLD ON HOMOGENEOUS SPACES

, 711

"... Abstract. We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λu be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ> λu then there is a. ..."

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Abstract. We consider the Poisson Boolean model of continuum percolation on a homogeneous space M. Let λ be the intensity of the underlying Poisson process. Let λu be the infimum of the set of intensities that a.s. produce a unique unbounded component. First we show that if λ> λu then there is a.s. a unique unbounded component at λ. Then we let M = H 2 × R and show that at λu there is a.s. not a unique unbounded component. These results are continuum analogies of theorems by Häggström, Peres and Schonmann. 1. Introduction and

### 1Universidade Federal do ABC

, 2014

"... We consider the Boolean discrete percolation model on graphs satisfying a dou-bling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percola-tion, provided that the retention parameter of t ..."

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We consider the Boolean discrete percolation model on graphs satisfying a dou-bling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percola-tion, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. An interesting example of such graphs is given by the Cayley graph of the discrete Heinsenberg group which can not be embedded in Rn for any n. Therefore, the absence of percolation in doubling graphs does not follow from the subcriticality of the Boolean percolation model in Rn and standard coupling arguments. Finally, we give sufficient conditions for ergodicity of the Boolean discrete percolation model.

### Non-uniqueness phase of Bernoulli percolation on reflection groups for some polyhedra in H3

, 2014

"... In the present paper I consider Cayley graphs of reflection groups of finite-sided Coxeter polyhedra in 3-dimensional hyperbolic space H3, with standard sets of generators. As the main result, I prove the existence of non-trivial non-uniqueness phase of bond and site Bernoulli percolation on such gr ..."

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In the present paper I consider Cayley graphs of reflection groups of finite-sided Coxeter polyhedra in 3-dimensional hyperbolic space H3, with standard sets of generators. As the main result, I prove the existence of non-trivial non-uniqueness phase of bond and site Bernoulli percolation on such graphs, i.e. that pc < pu, for two classes of such polyhedra: • for any k-hedra as above with k ≥ 13; • for any compact right-angled polyhedra as above. I also establish a natural lower bound for the growth rate of such Cayley graphs (when the number of faces of the polyhedron is ≥ 6; see thm. 5.2) and an upper bound for the growth rate of the sequence ( # {simple cycles of length n through o})n for a regular graph of degree ≥ 2 with a distinguished vertex o, depending on its spectral radius (see thm. 5.1 and rem. 2.3), both used to prove the main result. 1

### Clusters in middle-phase percolation on hyperbolic plane

, 2013

"... I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one end) and on their duals. It is known from [BS01] that in such a graph G we have three essential phases ..."

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I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one end) and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i. e. 0 < pc(G) < pu(G) < 1, where pc is the critical probability and pu – the unification probability. I prove that in the middle phase a. s. all the ends of all the infinite clusters have one-point boundary in ∂H2. This result is similar to some results in [Lal]. 1