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


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.


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

Mathematics Subject Classification (2000)

Primary 60K35 82B43 



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.


  1. 1.
    Chandler R., Koplick J., Lerman K., Willemsen J.F.: Capillary displacement and percolation in porous media. J. Fluid Mech. 119, 249–267 (1982)MATHCrossRefGoogle Scholar
  2. 2.
    Chayes J.T., Chayes L., Frölich J.: The low-temperature behavior of disordered magnets. Commun. Math. Phys. 100, 399–437 (1985)CrossRefGoogle Scholar
  3. 3.
    Chayes J.T., Chayes L., Newman C.: The stochastic geometry of invasion percolation. Commun. Math. Phys. 101, 383–407 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Damron M., Sapozhnikov A., Vágvölgyi B.: Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37, 2297–2331 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Damron, M., Sapozhnikov, A.: arXiv:0903.4496 (2009)Google Scholar
  6. 6.
    Diestel R.: Graph Theory, 2nd edn. Springer, New York (2000)Google Scholar
  7. 7.
    Garban, C., Pete, G.: Personal communication (2009)Google Scholar
  8. 8.
    Goodman, J.: Exponential growth of ponds for invasion percolation on regular trees (2009, preprint)Google Scholar
  9. 9.
    Grimmett G.: Percolation, 2nd edn. Springer, Berlin (1999)MATHGoogle Scholar
  10. 10.
    Járai A.A.: Invasion percolation and the incipient infinite cluster in 2D. Commun. Math. Phys. 236, 311–334 (2003)MATHCrossRefGoogle Scholar
  11. 11.
    Kesten, H.: A scaling relation at criticality for 2D-percolation. Percolation theory and ergodic theory of infinite particle systems (Minneapolis, Minn., 1984–1985), IMA Vol. Math. Appl., vol. 8, pp. 203–212. Springer, New York (1987)Google Scholar
  12. 12.
    Kesten H.: The incipient infinite cluster in two-dimesional percolation. Probab. Theory Rel. Fields 73, 369–394 (1986)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Kesten H.: Scaling relations for 2D percolation. Commun. Math. Phys. 109, 109–156 (1987)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Lenormand R., Bories S.: Description d’un mecanisme de connexion de liaision destine a l’etude du drainage avec piegeage en milieu poreux. C. R. Acad. Sci. 291, 279–282 (1980)Google Scholar
  15. 15.
    Nagaev S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789 (1979)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Newman C., Stein D.L.: Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E. 51, 5228–5238 (1995)CrossRefGoogle Scholar
  17. 17.
    Nguyen, B.G.: Correlation lengths for percolation processes. Ph. D. Thesis, University of California, Los Angeles (1985)Google Scholar
  18. 18.
    Nolin P.: Near critical percolation in two-dimensions. Electron. J. Probab. 13, 1562–1623 (2008)MathSciNetMATHGoogle Scholar
  19. 19.
    Reimer D.: Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput. 9, 27–32 (2000)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    van den Berg J., Járai A.A., Vágvölgyi B.: The size of a pond in 2D invasion percolation. Electron. Comm. Probab. 12, 411–420 (2007)MathSciNetMATHGoogle Scholar
  21. 21.
    Werner, W.: Lectures on two-dimensional critical percolation. arXiv: 0710.0856 (2007)Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

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

Personalised recommendations