Hyperscaling for Oriented Percolation in $$1+1$$ 1 + 1 Space–Time Dimensions

Percolation in ∞ + 1 Dimensions

We investigate percolation on the graph of the direct product T × Z of a regular tree T and the line Z, in which each 'tree' edge is open with probability τ and each 'line' edge with probability λ. There are three non-trivial phases, corresponding to the existence of 0, ∞, and 1 infinite open clusters. Such results may be obtained also for the graph T × Z d where d ≥ 2.

Oriented Percolation in Two Dimensions

Percolation in 1 + 1 Dimensions at the Uniqueness Threshold

For independent density p site percolation on the (transitive non-amenable) graph T b Z, where T b is a homogeneous tree of degree b + 1, and b is supposed to be large, it is shown that for p = p u = inffp: a.s. there is a unique innnite clusterg there are a.s. innnitely many inn-nite clusters. This contrasts with a recent result of Benjamini and Schramm, according to whom for transitive non-am...

Construction of the incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions

We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on Zd × Z+, for d + 1 > 4 + 1. We consider two different constructions. For the first construction, we define Pn(E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x, n) ∈ Z×Z+, summing this probability over x ∈ Zd, and normalisin...

The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction

We consider critical spread-out oriented percolation above 4+1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/Bn2 as n →∞, where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the ...