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28
Percolation perturbations in potential theory and random walks
, 1998
"... We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which ..."
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Cited by 37 (14 self)
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We show that on a Cayley graph of a nonamenable group, a.s. the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph a.s. admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting nonconstant harmonic Dirichlet functions, a.s. the infinite clusters of pBernoulli percolation also have nonconstant harmonic Dirichlet functions when p is sufficiently close to 1. Many conjectures and questions are presented.
A bird’seye view of uniform spanning trees and forests
 Microsurveys in Discrete Probability, volume 41 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1998
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Gromov hyperbolicity of Riemann surfaces
"... Abstract. In this paper we study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its “building block components”. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some g ..."
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Cited by 8 (3 self)
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Abstract. In this paper we study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its “building block components”. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated to it. These results clarify how the decomposition of a Riemann surface in Ypieces and funnels affects on the hyperbolicity of the surface. The results simplify the topology of the surface and allow to obtain global results from local information. 1. Introduction. A good way to understand the important connections between graphs and Potential Theory on Riemannian manifolds (see e.g. [APR], [ARY], [CFPR], [FR2], [HS], [K1], [K2], [K3], [R1], [R2], [So]) is to study Gromov hyperbolic spaces. This approach allows us to establish a general setting to work simultaneously with graphs and manifolds, in the context of metric spaces. Besides, the idea of Gromov hyperbolicity grasps the essence
GRAPHS OF BOUNDED DEGREE AND THE pHARMONIC BOUNDARY
, 806
"... Abstract. Let p be a real number greater than one and let G be a connected graph of bounded degree. In this paper we introduce the pharmonic boundary of G. We use this boundary to characterize the graphs G for which the constant functions are the only pharmonic functions on G. It is shown that any ..."
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Cited by 8 (3 self)
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Abstract. Let p be a real number greater than one and let G be a connected graph of bounded degree. In this paper we introduce the pharmonic boundary of G. We use this boundary to characterize the graphs G for which the constant functions are the only pharmonic functions on G. It is shown that any continuous function on the pharmonic boundary of G can be extended to a function that is pharmonic on G. Some properties of this boundary that are preserved under roughisometries are also given. Now let Γ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced ℓ pcohomology of Γ in terms of the cardinality of its pharmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on Γ, the pharmonic boundary of Γ and the first reduced ℓ pcohomology of Γ. 1.
A characterization of Gromov hyperbolicity of surfaces with variable negative curvature
"... Abstract. In this paper we show that, in order to check Gromov hyperbolicity of any surface with curvature K ≤ −k2 < 0, we just need to verify the Rips condition on a very small class of triangles, namely, those contained in simple closed geodesics. This result is, in fact, a new characterization ..."
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Cited by 5 (3 self)
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Abstract. In this paper we show that, in order to check Gromov hyperbolicity of any surface with curvature K ≤ −k2 < 0, we just need to verify the Rips condition on a very small class of triangles, namely, those contained in simple closed geodesics. This result is, in fact, a new characterization of Gromov hyperbolicity for this kind of surfaces.
A real variable characterization of Gromov hyperbolicity of flute surfaces
, 2007
"... Abstract. In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This ..."
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Cited by 5 (5 self)
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Abstract. In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasiisometric.