Abstract:
We establish two links between two-dimensional invasion percolation and Kesten's incipient infinite cluster (IIC). We first prove that the k th moment of the number of invaded sites within the box [−n, n]×[−n, n] is of order (n 2π n )k, for k≥1, where π n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n. This improves a result of Y. Zhang. We show that the size of the invaded region, when scaled by n 2π n , is tight.
Secondly, we prove that the invasion cluster looks asymptotically like the IIC, when viewed from an invaded site v, in the limit |v|→∞. We also establish this when an invaded site v is chosen at random from a box of radius n, and n→∞.
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Received: 3 December 2000 / Accepted: 3 December 2002 Published online: 18 February 2003
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ID="⋆" Present address: CWI, PNA 3, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail:jarai@cwi.nl
Communicated by M. Aizenman
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Járai, A. Invasion Percolation and the Incipient Infinite Cluster in 2D. Commun. Math. Phys. 236, 311–334 (2003). https://doi.org/10.1007/s00220-003-0796-6
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DOI: https://doi.org/10.1007/s00220-003-0796-6