Probability Theory and Related Fields

, Volume 150, Issue 1, pp 257–294

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

Authors

    • Mathematics DepartmentPrinceton University
  • Artëm Sapozhnikov
    • EURANDOM
Open AccessArticle

DOI: 10.1007/s00440-010-0274-y

Cite this article as:
Damron, M. & Sapozhnikov, A. Probab. Theory Relat. Fields (2011) 150: 257. doi:10.1007/s00440-010-0274-y

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(2 n ) 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.

Keywords

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

Mathematics Subject Classification (2000)

Primary 60K35 82B43

Copyright information

© The Author(s) 2010