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Directed percolation and random walk. (English) Zbl 1010.60087
Sidoravicius, Vladas (ed.), In and out of equilibrium. Probability with a physics flavor. Papers from the 4th Brazilian school of probability, Mambucaba, Brazil, August 14-19, 2000. Boston: Birkhäuser. Prog. Probab. 51, 273-297 (2002).
The authors extend ‘dynamic renormalization’ techniques, developed earlier for undirected percolation and the contact model, to the setting of directed percolation on $$\mathbb{Z}^d$$ for $$d\geq 2$$. The authors show that the percolation probability at the critical level vanishes. Another new result is a type of uniqueness theorem: for every pair $$x$$ and $$y$$ of vertices which lie in infinite open paths, there exists almost surely a third vertex $$z$$ which is joined to infinity and which is attainable from $$x$$ and $$y$$ along directed open paths. The authors also prove in the supercritical case that a random walk on an infinite directed cluster is transient, almost surely, when $$d\geq 3$$. It is shown how to adapt the block arguments to systems with infinite range, subject to certain conditions on the edge probabilities.
For the entire collection see [Zbl 0996.00040].

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60K37 Processes in random environments 82B43 Percolation 60G50 Sums of independent random variables; random walks
##### Keywords:
directed percolation; renormalization; infinite cluster
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