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158
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys
, 2005
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Random walk on the incipient infinite cluster on trees
 Illinois J. Math
"... Abstract. Let G be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree n0 + 1. We obtain estimates for the transition density of the the continuous time simple random walk Y on G; the process satisfies anomalous diffusion and has spectral dimension 4 ..."
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Cited by 61 (16 self)
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Abstract. Let G be the incipient infinite cluster (IIC) for percolation on a homogeneous tree of degree n0 + 1. We obtain estimates for the transition density of the the continuous time simple random walk Y on G; the process satisfies anomalous diffusion and has spectral dimension 4
Percolation in the Hyperbolic Plane
, 2000
"... Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation ..."
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Cited by 47 (4 self)
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Following is a study of percolation in the hyperbolic plane H 2 and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and PoissonVoronoiBernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, p ∈ (0, pc], there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, p ∈ (pc, pu), there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, p∈[pu, 1), there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of pc in the PoissonVoronoiBernoulli percolation process on the intensity of the underlying Poisson process.
Indistinguishability of percolation clusters
 Ann. Probab
, 1999
"... Abstract. We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long ..."
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Cited by 46 (14 self)
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Abstract. We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. longrange order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products, and inequalities for pu. §1. Introduction. Grimmett and Newman (1990) showed that if T is a regular tree of sufficiently high degree, then there are p ∈ (0, 1) such that Bernoulli(p) percolation on T × Z has infinitely many infinite components a.s. Benjamini and Schramm (1996) conjectured that the same is true for any Cayley graph of any finitely generated nonamenable group. (A finitely
Combinatorial Bandits
"... We study sequential prediction problems in which, at each time instance, the forecaster chooses a binary vector from a certain fixed set S ⊆ {0, 1} d and suffers a loss that is the sum of the losses of those vector components that equal to one. The goal of the forecaster is to achieve that, in the l ..."
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Cited by 46 (6 self)
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We study sequential prediction problems in which, at each time instance, the forecaster chooses a binary vector from a certain fixed set S ⊆ {0, 1} d and suffers a loss that is the sum of the losses of those vector components that equal to one. The goal of the forecaster is to achieve that, in the long run, the accumulated loss is not much larger than that of the best possible vector in the class. We consider the “bandit ” setting in which the forecaster has only access to the losses of the chosen vectors. We introduce a new general forecaster achieving a regret bound that, for a variety of concrete choices of S, is of order √ nd ln S  where n is the time horizon. This is not improvable in general and is better than previously known bounds. We also point out that computationally efficient implementations for various interesting choices of S exist. 1
Scaling limits for minimal and random spanning trees in two dimensions
, 1998
"... A general formulation is presented for continuum scaling limits of stochastic spanning trees. Tightness of the distribution, as δ → 0, is established for the following twodimensional examples: the uniformly random spanning tree on δZ 2, the minimal spanning tree on δZ 2 (with random edge lengths), a ..."
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Cited by 41 (8 self)
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A general formulation is presented for continuum scaling limits of stochastic spanning trees. Tightness of the distribution, as δ → 0, is established for the following twodimensional examples: the uniformly random spanning tree on δZ 2, the minimal spanning tree on δZ 2 (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in R 2 with density δ −2. A continuum limit is expressed through a consistent collection of trees (made of curves) which includes a spanning tree for every finite set of points in the plane. Sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for δ> 0), ii) the tree branches are given by curves which are regular in the sense of Hölder continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of R², of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in R 2, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scaleinvariant power bounds on the probabilities of repeated crossings of annuli.
Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness
 Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten
, 1999
"... Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter pc ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasitransitive, there is another critical value pu ∈ [pc, 1] such that the nu ..."
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Cited by 39 (9 self)
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Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter pc ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasitransitive, there is another critical value pu ∈ [pc, 1] such that the number of infinite clusters is a.s. ∞ for p ∈ (pc, pu), and a.s. one for p> pu. We prove a simultaneous version of this result in the canonical coupling of the percolation processes for all p ∈ [0, 1]. Simultaneously for all p ∈ (pc, pu), we also prove that each infinite cluster has uncountably many ends. For p> pc we prove that all infinite clusters are indistinguishable by robust properties. Under the additional assumption that G is unimodular, we prove that a.s. for all p1 < p2 in (pc, pu), every infinite cluster at level p2 contains infinitely many infinite clusters at level p1. We also show that any Cartesian product G of d infinite connected graphs of bounded degree satisfies pu(G) ≤ pc(Z d).
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.
Probability on trees: an introductory climb.
 Ecole d'été de probabilité de SaintFlour XXVIII, number 1717 in L.N. in Math.,
, 1999
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