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Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters
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  • Published: 11 March 2010

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

  • Michael Damron1 &
  • Artëm Sapozhnikov2 

Probability Theory and Related Fields volume 150, pages 257–294 (2011)Cite this article

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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 p c . 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.

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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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Authors and Affiliations

  1. Mathematics Department, Princeton University, Fine Hall, Washington Rd., Princeton, NJ, 08544, USA

    Michael Damron

  2. EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Artëm Sapozhnikov

Authors
  1. Michael Damron
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  2. Artëm Sapozhnikov
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Corresponding author

Correspondence to Michael Damron.

Additional information

Research funded by NSF grant OISE-0730136 and an NSF postdoctoral fellowship.

Research partially supported by the Netherlands Organisation for Scientific Research (NWO) under grant number 613.000.429.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Damron, M., Sapozhnikov, A. Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theory Relat. Fields 150, 257–294 (2011). https://doi.org/10.1007/s00440-010-0274-y

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  • Received: 14 November 2009

  • Revised: 25 January 2010

  • Published: 11 March 2010

  • Issue Date: June 2011

  • DOI: https://doi.org/10.1007/s00440-010-0274-y

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Keywords

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

Mathematics Subject Classification (2000)

  • Primary 60K35
  • 82B43
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