Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

Abstract:

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ^d × 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^d × ℤ₊, summing this probability over x  ℤ ^d, 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.