One dimensional \(1/| j-i| ^ s\) percolation models: The existence of a transition for s\(\leq 2\). (English) Zbl 0604.60097
Consider a one-dimensional independent bond percolation model with \(p_ j\) denoting the probability of an occupied bond between integer sites i and \(i\pm j\), \(j\geq 1\). If \(p_ j\) is fixed for \(j\geq 2\) and \(\lim_{j\to \infty}j^ 2p_ j>1\), 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 \(\lim_{j\to \infty}j^ sp_ j>0\) for some \(s<2\). Analogous results are valid for one-dimensional site-bond percolation models.
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 \(\lim_{j\to \infty}j^ sp_ j>0\) for some \(s<2\). Analogous results are valid for one-dimensional site-bond percolation models.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
Keywords:
bond percolation model; one-dimensional Ising models; renormalization group; Oriented percolationReferences:
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