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November 2009 Relations between invasion percolation and critical percolation in two dimensions
Michael Damron, Artëm Sapozhnikov, Bálint Vágvölgyi
Ann. Probab. 37(6): 2297-2331 (November 2009). DOI: 10.1214/09-AOP462


We study invasion percolation in two dimensions. We compare connectivity properties of the origin’s invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k≥1, the k-point function of the first so-called pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint pc-open clusters. Further, for k>1, we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular.


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Michael Damron. Artëm Sapozhnikov. Bálint Vágvölgyi. "Relations between invasion percolation and critical percolation in two dimensions." Ann. Probab. 37 (6) 2297 - 2331, November 2009.


Published: November 2009
First available in Project Euclid: 16 November 2009

zbMATH: 1247.60134
MathSciNet: MR2573559
Digital Object Identifier: 10.1214/09-AOP462

Primary: 60K35 , 82B43

Keywords: correlation length , Critical percolation , Incipient infinite cluster , Invasion percolation , invasion ponds , Near-critical percolation , scaling relations , singularity

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 6 • November 2009
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