Construction of the incipient infinite cluster for spread-out oriented percolation above 4 + 1 dimensions Remco van der Hofstad ∗

We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out ori-ented 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) ∈ Zd×Z+, summing this proba-bility over x ∈ Zd, and normalising the sum to get a probability measure. We let n→ ∞ and prove existence of a limiting measure P∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n−1, we prove existence of a limiting measure Q∞, with Q ∞ = P∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under P∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+ 1> 4 + 1. 1 Introduction an