Results 1  10
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Bootstrap percolation on infinite trees and nonamenable groups
 Combinatorics, Probability and Computing
"... Abstract. Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time ..."
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Cited by 48 (6 self)
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step, then it becomes occupied in the next step. This process is wellstudied on Z d; here we investigate it on regular and general infinite trees and on nonamenable Cayley graphs. The critical probability is the infimum of those values of p for which the process achieves complete occupation
The nonamenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups
, 2002
"... Abstract. We show that if H is a quasiconvex subgroup of infinite index in a nonelementary hyperbolic group G then the Schreier coset graph for G relative to H is nonamenable. 1. ..."
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Cited by 7 (2 self)
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Abstract. We show that if H is a quasiconvex subgroup of infinite index in a nonelementary hyperbolic group G then the Schreier coset graph for G relative to H is nonamenable. 1.
GroupInvariant Percolation on Graphs
 Geom. Funct. Anal
, 1999
"... . Let G be a closed group of automorphisms of a graph X . We relate geometric properties of G and X , such as amenability and unimodularity, to properties of Ginvariant percolation processes on X , such as the number of infinite components, the expected degree, and the topology of the components. O ..."
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Cited by 120 (40 self)
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Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group. We show that G is amenable iff for all ff ! 1, there is a Ginvariant site percolation process ! on X with P[x 2 !] ? ff
Nazarov: Perfect matchings as IID factors on nonamenable groups
 Europ. J. Combin
, 2011
"... ar ..."
spectral radius and percolation constants on nonamenable Cayley graphs., arXiv preprint arXiv:1206.2183
"... Abstract. Motivated by the BenjaminiSchramm nonunicity of percolation conjecture we study the following question. For a given finitely generated nonamenable group Γ, does there exist a generating set S such that the Cayley graph (Γ, S), without loops and multiple edges, has nonunique percolati ..."
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Cited by 4 (1 self)
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unique percolation, i.e., pc(Γ, S) < pu(Γ, S)? We show that this is true if Γ contains an infinite normal subgroup N such that Γ/N is nonamenable. Moreover for any finitely generated group G containing Γ there exists a generating set S ′ of G such that pc(G,S′) < pu(G,S′). In particular this applies to free
Uniform nonamenability, cost, and the first ℓ2Betti number
 Groups Geom. Dyn
, 2008
"... Abstract. It is shown that 2β1(Γ) ≤ h(Γ) for any countable group Γ, where β1(Γ) is the first ℓ 2Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with nonvanishing first ℓ 2Betti number is uniformly nonamenable. We then define isoperimetric constants in ..."
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Cited by 2 (2 self)
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Abstract. It is shown that 2β1(Γ) ≤ h(Γ) for any countable group Γ, where β1(Γ) is the first ℓ 2Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with nonvanishing first ℓ 2Betti number is uniformly nonamenable. We then define isoperimetric constants
UNIFORM NONAMENABILITY AND THE FIRST ℓ 2BETTI NUMBER
, 711
"... Abstract. It is shown that 2β1(Γ) ≤ h(Γ) for any countable group Γ, where β1(Γ) is the first ℓ 2Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with nonvanishing first ℓ 2Betti number is uniformly nonamenable. We then define isoperimetric constants in ..."
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Abstract. It is shown that 2β1(Γ) ≤ h(Γ) for any countable group Γ, where β1(Γ) is the first ℓ 2Betti number and h(Γ) the uniform isoperimetric constant. In particular, a countable group with nonvanishing first ℓ 2Betti number is uniformly nonamenable. We then define isoperimetric constants
ControlFlow Analysis of HigherOrder Languages
, 1991
"... representing the official policies, either expressed or implied, of ONR or the U.S. Government. Keywords: dataflow analysis, Scheme, LISP, ML, CPS, type recovery, higherorder functions, functional programming, optimising compilers, denotational semantics, nonstandard Programs written in powerful, ..."
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Cited by 362 (10 self)
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at compile time. I give examples of how this information can be used to perform several dataflow analysis optimisations, including copy propagation, inductionvariable elimination, uselessvariable elimination, and type recovery. The analysis is defined in terms of a nonstandard semantic interpretation
Results 1  10
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3,962