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


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.


Neural Network Nonlinear Dynamics Long Range Renormalization Group Variable Size 
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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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