Directed percolation and random walk. (English) Zbl 1010.60087

Grimmett, Geoffrey; Hiemer, Philipp

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].

Reviewer: Ilya S.Molchanov (Glasgow)

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