Directed Percolation and Random Walk

Abstract

Abstract. 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 Z d 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.

The survival probability for critical spread-out oriented percolation above 4+14+1 dimensions. II. Expansion

“…This requires that y = 0 and M = 0, and contributes at most CC 3 θ (n + 1) −3 . With this observation, we can improve (8.49) to (4) In this section, we prove the bound (8.7) on the error term e (N ) n+1 (4 (4,2) in Section 8.6.3. The proof of (8.7) is completed at the end of Section 8.6.3.…”

“…In the oriented setting, it is known that there is no percolation at the critical threshold p = p c [2,4], so that lim n→∞ θ n (p c ) = 0. Our goal is to study the manner in which θ n (p c ) tends to zero as n → ∞ when d > 4.…”

“…General results of [2,4] imply that lim n→∞ θ n (p c ) = 0. For the spread-out model in dimension d > 4, with L sufficiently large, the same conclusion was shown in [1] to follow from the triangle condition.…”

“…In Section 8.2, we provide some preliminary convolution bounds. Then we estimate the five error terms in order of increasing difficulty, namely e (N ) n+1 (3), e (N ) n+1 (5), e (N ) n+1 (2), e (N ) n+1 (4), e (N ) n+1 (1), in Sections 8.3-8.7, respectively.…”

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“…For the nearest-neighbor CP and OP with d ≥ 1 and for the spread-out OP with d ≥ 1 and L ≥ 1, it has been proved [5,10] that there exists a critical point λ c ∈ (0, ∞) (λ c ∈ (0, |Ω|) for OP) such that θ(λ) = 0 when and only when λ ≤ λ c . Therefore Θ t (λ) = θ t (λ) when λ ≤ λ c .…”