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

C. M. Newman, L. S. Schulman

Research output: Contribution to journalArticlepeer-review


Consider a one-dimensional independent bond percolation model with pj denoting the probability of an occupied bond between integer sites i and i±j, j≧1. If pj is fixed for j≧2 and {Mathematical expression}j2pj>1, then (unoriented) percolation occurs for p1 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 for p1 close to 1 if {Mathematical expression}jspj>0 for some s<2. Analogous results are valid for one-dimensional site-bond percolation models.

Original languageEnglish (US)
Pages (from-to)547-571
Number of pages25
JournalCommunications In Mathematical Physics
Issue number4
StatePublished - Dec 1986

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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