The Annals of Probability

Relations between invasion percolation and critical percolation in two dimensions

Michael Damron, Artëm Sapozhnikov, and Bálint Vágvölgyi

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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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Ann. Probab. Volume 37, Number 6 (2009), 2297-2331.

First available in Project Euclid: 16 November 2009

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Invasion percolation invasion ponds critical percolation near-critical percolation correlation length scaling relations incipient infinite cluster singularity


Damron, Michael; Sapozhnikov, Artëm; Vágvölgyi, Bálint. Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37 (2009), no. 6, 2297--2331. doi:10.1214/09-AOP462.

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