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

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 {Mathematical expression}j²p_j>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}j^sp_j>0 for some s<2. Analogous results are valid for one-dimensional site-bond percolation models.