2001/08/08 by Geoffrey Grimmett, Grimmett, Geoffrey, Philipp Hiemer +1
Mathematics · Physics and Astronomy · #60G50 #60K35 #60K37 #82B43 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60G50 #msc:60K35 #msc:60K37 #msc:82B43
paper · pdf · doi:10.48550/arxiv.math/0108062
arxiv created 2001/08/08 · openalex publication_date 2001/08/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 where d ≥ 2. The first 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. Secondly, it is proved that a random walk on an infinite directed cluster is transient, almost surely, when d ≥ 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.