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) 2 ZdZ+, summing this proba-bility over x 2 Zd, and normalising the sum to get a probability measure. We let n!1 and prove existence of a limiting measure P1, 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! 1. Under the assumption that the critical survival probability is asymptotic to a multiple of n−1, we prove existence of a limiting measure Q1, with Q1 = P1. 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 P1. 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