Directed percolation and random walk

Techniques of ‘dynamic renormalization’, developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on ℤd whered ≥ 2.The first new result is a type of uniqueness theorem: for every pairxand 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 fromxand y along directed open paths. Secondly, it is proved in the supercritical case that a random walk on an infinite directed cluster is transient, almost surely, whend ℤ 3.And finally, the block arguments of the paper may be adapted to systems with infinite range, subject to certain conditions on the edge probabilities.

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