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Communications in Mathematical Physics

, Volume 104, Issue 4, pp 547–571 | Cite as

One dimensional 1/|j − i| S percolation models: The existence of a transition forS≦2

  • C. M. Newman
  • L. S. Schulman
Article

Abstract

Consider a one-dimensional independent bond percolation model withpj denoting the probability of an occupied bond between integer sitesi andi±j,j≧1. Ifpj is fixed forj≧2 and\(\mathop {\lim }\limits_{j \to \infty }\)j2pj>1, then (unoriented) percolation occurs forp1 sufficiently close to 1. This result, analogous to the existence of spontaneous magnetization in long range one-dimensional Ising models, is proved by an inductive series of bounds based on a renormalization group approach using blocks of variable size. Oriented percolation is shown to occur forp1 close to 1 if\(\mathop {\lim }\limits_{j \to \infty }\)jspj>0 for somes<2. Analogous results are valid for one-dimensional site-bond percolation models.

Keywords

Neural Network Nonlinear Dynamics Long Range Renormalization Group Variable Size 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • C. M. Newman
    • 1
  • L. S. Schulman
    • 1
    • 2
    • 3
  1. 1.Department of Mathematics and Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of Physics, TechnionHaifaIsrael
  3. 3.Department of PhysicsClarkson UniversityPotsdamUSA

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