Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

Berger, Noam

doi:10.1007/s002200200617

Cited by 88 publications

(121 citation statements)

References 20 publications

(39 reference statements)

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“…This is compatible with our results, which imply that there is no infinite cluster at criticality for d > 3α (and here α = 1). Berger [12] In a recent paper, Chen and Sakai [18] study oriented percolation in the spread-out power-law case. Using similar methods, they prove that the two-point function in oriented percolation obeys an infrared bound if d > 2(α ∧ 2), which implies mean-field behavior of the model.…”

Section: Discussion and Related Workmentioning

confidence: 99%

Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk

2008

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We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if d > 2(α ∧ 2) for self-avoiding walk and the Ising model, and d > 3(α ∧ 2) for percolation, where d denotes the dimension and α the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. [14].MSC 2000. 82B41, 82B43, 60K35.

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Section: Discussion and Related Workmentioning

confidence: 99%

Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk

2008

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“…Here we conjecture: Note that, according to this conjecture, in d = 1, the interval α ∈ (0, 2) of "interesting" exponents is larger than the interval for which an infinite connected component may occur even without the "help" of nearest neighbor connections. On the other hand, in dimensions d ≥ 3, the interval conjectured for stable convergence is strictly smaller than that of "genuine" long-range percolation behavior, as defined, e.g., in terms of the scaling of graph distance with Euclidean distance; cf [4,5,8].…”

Section: B Some Questions and Conjecturesmentioning

confidence: 98%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

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193

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Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

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“…This establishes parts (2) and (3) of the theorem. It thus remains to prove the strict inequality between x 1 and x 2 = · · · = x 1 in part (1)-the rest follows by Lemma 5.4(1)-and the properties of β → h 1 (β) in part (4).…”

Section: Potts Model: Positive Fieldsmentioning

confidence: 99%

Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

Biskup¹,

Chayes²,

Crawford³

2006

J Stat Phys

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We consider a class of spin systems on Z d with vector valued spins (S x ) that interact via the pair-potentials J x,y S x · S y . The interactions are generally spreadout in the sense that the J x,y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d ≥ 3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1, 2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique "state," then in any sequence of translation-invariant Gibbs states various observables converge to their meanfield values and the states themselves converge to a product measure.

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Transience, Recurrence and Critical Behavior¶for Long-Range Percolation

Cited by 88 publications

References 20 publications

Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk

Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk

Quenched invariance principle for simple random walk on percolation clusters

Mean-Field Driven First-Order Phase Transitions in Systems with Long-Range Interactions

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