The Annals of Probability

Invasion percolation on regular trees

Omer Angel, Jesse Goodman, Frank den Hollander, and Gordon Slade

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Abstract

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its r-point function for any r≥2 and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally.

A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

Article information

Source
Ann. Probab., Volume 36, Number 2 (2008), 420-466.

Dates
First available in Project Euclid: 29 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1204306958

Digital Object Identifier
doi:10.1214/07-AOP346

Mathematical Reviews number (MathSciNet)
MR2393988

Zentralblatt MATH identifier
1145.60050

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

Keywords
Invasion percolation cluster incipient infinite cluster r-point function cluster size simple random walk Poisson point process

Citation

Angel, Omer; Goodman, Jesse; den Hollander, Frank; Slade, Gordon. Invasion percolation on regular trees. Ann. Probab. 36 (2008), no. 2, 420--466. doi:10.1214/07-AOP346. https://projecteuclid.org/euclid.aop/1204306958


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