Abstract:
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) ℤd × 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.
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Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002
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ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl
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van der Hofstad, R., den Hollander, F. & Slade, G. Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions. Commun. Math. Phys. 231, 435–461 (2002). https://doi.org/10.1007/s00220-002-0728-x
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DOI: https://doi.org/10.1007/s00220-002-0728-x