One dimensional 1/| j - i| S percolation models: The existence of a transition for S≦2
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 andmathop {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 ifmathop {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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- December 1986
- DOI:
- 10.1007/BF01211064
- Bibcode:
- 1986CMaPh.104..547N