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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References
Chandler R., Koplick J., Lerman K., Willemsen J.F.: Capillary displacement and percolation in porous media. J. Fluid Mech. 119, 249–267 (1982)
Chayes J.T., Chayes L., Frölich J.: The low-temperature behavior of disordered magnets. Commun. Math. Phys. 100, 399–437 (1985)
Chayes J.T., Chayes L., Newman C.: The stochastic geometry of invasion percolation. Commun. Math. Phys. 101, 383–407 (1985)
Damron M., Sapozhnikov A., Vágvölgyi B.: Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37, 2297–2331 (2009)
Damron, M., Sapozhnikov, A.: arXiv:0903.4496 (2009)
Diestel R.: Graph Theory, 2nd edn. Springer, New York (2000)
Garban, C., Pete, G.: Personal communication (2009)
Goodman, J.: Exponential growth of ponds for invasion percolation on regular trees (2009, preprint)
Grimmett G.: Percolation, 2nd edn. Springer, Berlin (1999)
Járai A.A.: Invasion percolation and the incipient infinite cluster in 2D. Commun. Math. Phys. 236, 311–334 (2003)
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)
Kesten H.: The incipient infinite cluster in two-dimesional percolation. Probab. Theory Rel. Fields 73, 369–394 (1986)
Kesten H.: Scaling relations for 2D percolation. Commun. Math. Phys. 109, 109–156 (1987)
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)
Nagaev S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789 (1979)
Newman C., Stein D.L.: Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E. 51, 5228–5238 (1995)
Nguyen, B.G.: Correlation lengths for percolation processes. Ph. D. Thesis, University of California, Los Angeles (1985)
Nolin P.: Near critical percolation in two-dimensions. Electron. J. Probab. 13, 1562–1623 (2008)
Reimer D.: Proof of the van den Berg–Kesten conjecture. Combin. Probab. Comput. 9, 27–32 (2000)
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)
Werner, W.: Lectures on two-dimensional critical percolation. arXiv: 0710.0856 (2007)
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.
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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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DOI: https://doi.org/10.1007/s00440-010-0274-y
Keywords
- Invasion percolation
- Invasion ponds
- Critical percolation
- Near critical percolation
- Correlation length
- Scaling relations
- Incipient infinite cluster