Probability Theory and Related Fields

, Volume 150, Issue 1–2, pp 257–294 | Cite as

Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

Open Access
Article

Abstract

We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IICs). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the distribution of the number of outlets in an annulus. This result leads to almost sure bounds for the number of outlets in a box B(2n) and for the decay rate of the weight of the kth outlet to pc. We then prove existence of multiple-armed IIC measures for any number of arms and for any color sequence which is alternating or monochromatic. We use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin.

Keywords

Invasion percolation Invasion ponds Critical percolation Near critical percolation Correlation length Scaling relations Incipient infinite cluster 

Mathematics Subject Classification (2000)

Primary 60K35 82B43 

Notes

Acknowledgments

We would like to thank C. Newman for suggesting some of these problems.We thank R. van den Berg and C. Newman for helpful discussions. We also thank G. Pete for discussions related to arm-separation statements for multiple-armed IICs.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.EURANDOMEindhovenThe Netherlands

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