# One dimensional 1/|*j − i*|^{ S } percolation models: The existence of a transition for*S*≦2

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## Abstract

Consider a one-dimensional independent bond percolation model with*p*_{j} denoting the probability of an occupied bond between integer sites*i* and*i*±*j*,*j*≧1. If*p*_{j} is fixed for*j*≧2 and\(\mathop {\lim }\limits_{j \to \infty }\)*j*^{2}*p*_{j}>1, then (unoriented) percolation occurs for*p*_{1} 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*p*_{1} close to 1 if\(\mathop {\lim }\limits_{j \to \infty }\)*j*^{s}*p*_{j}>0 for some*s*<2. Analogous results are valid for one-dimensional site-bond percolation models.

## Keywords

Neural Network Nonlinear Dynamics Long Range Renormalization Group Variable Size## Preview

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