Relations between invasion percolation and critical percolation in two dimensions
Michael Damron, Artëm Sapozhnikov, and Bálint Vágvölgyi
Source: Ann. Probab. Volume 37, Number 6
(2009), 2297-2331.
Abstract
We study invasion percolation in two dimensions. We compare connectivity properties of the origin’s invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k≥1, the k-point function of the first so-called pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint pc-open clusters. Further, for k>1, we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular.
First Page:
Show
Hide
Keywords: Invasion percolation; invasion ponds; critical percolation; near-critical percolation; correlation length; scaling relations; incipient infinite cluster; singularity
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380790
Digital Object Identifier: doi:10.1214/09-AOP462
Zentralblatt MATH identifier: 05708802
Mathematical Reviews number (MathSciNet): MR2573559
References
[1] Angel, O., Goodman, J., den Hollander, F. and Slade, G. (2008). Invasion percolation on regular trees. Ann. Probab. 36 420–466.
Mathematical Reviews (MathSciNet): MR2393988
Zentralblatt MATH: 1145.60050
Digital Object Identifier: doi:10.1214/07-AOP346
Project Euclid: euclid.aop/1204306958
[2] Chandler, R., Koplick, J., Lerman, K. and Willemsen, J. F. (1982). Capillary displacement and percolation in porous media. J. Fluid Mech. 119 249–267.
[3] Chayes, J. T., Chayes, L. and Fröhlich, J. (1985). The low-temperature behavior of disordered magnets. Comm. Math. Phys. 100 399–437.
Mathematical Reviews (MathSciNet): MR802552
Digital Object Identifier: doi:10.1007/BF01206137
Project Euclid: euclid.cmp/1104113922
[4] Chayes, J. T., Chayes, L. and Newman, C. (1985). The stochastic geometry of invasion percolation. Comm. Math. Phys. 101 383–407.
Mathematical Reviews (MathSciNet): MR815191
Digital Object Identifier: doi:10.1007/BF01216096
Project Euclid: euclid.cmp/1104114182
[5] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1707339
[6] Járai, A. A. (2003). Invasion percolation and the incipient infinite cluster in 2D. Comm. Math. Phys. 236 311–334.
[7] Kesten, H. (1987). A scaling relation at criticality for 2D-percolation. In Percolation Theory and Ergodic Theory of Infinite Particle Systems (Minneapolis, Minn., 1984–1985). The IMA Volumes in Mathematics and its Applications 8 203–212. Springer, New York.
Mathematical Reviews (MathSciNet): MR894537
[8] Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369–394.
Mathematical Reviews (MathSciNet): MR859839
Zentralblatt MATH: 0584.60098
Digital Object Identifier: doi:10.1007/BF00776239
[9] Kesten, H. (1987). Scaling relations for 2D-percolation. Comm. Math. Phys. 109 109–156.
Mathematical Reviews (MathSciNet): MR879034
Digital Object Identifier: doi:10.1007/BF01205674
Project Euclid: euclid.cmp/1104116714
[10] Kunz, H. and Souillard, B. (1978). Essential singularity in percolation problems and asymptotic behavior of cluster size distribution. J. Statist. Phys. 19 77–106.
Mathematical Reviews (MathSciNet): MR496290
Digital Object Identifier: doi:10.1007/BF01020335
[11] Lenormand, R. and Bories, S. (1980). Description d’un mecanisme de connexion de liaision destine a l’etude du drainage avec piegeage en milieu poreux. C. R. Math. Acad. Sci. 291 279–282.
[12] Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1180522
[13] Nash, S. W. (1954). An extension of the Borel–Cantelli lemma. Ann. Math. Statist. 25 165–167.
Mathematical Reviews (MathSciNet): MR61777
Digital Object Identifier: doi:10.1214/aoms/1177728858
Project Euclid: euclid.aoms/1177728858
[14] Newman, C. M. and Schulman, L. S. (1981). Infinite clusters in percolation models. J. Statist. Phys. 26 613–628.
Mathematical Reviews (MathSciNet): MR648202
Zentralblatt MATH: 0509.60095
Digital Object Identifier: doi:10.1007/BF01011437
[15] Newman, C. and Stein, D. L. (1995). Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E. 51 5228–5238.
[16] Nguyen, B. G. (1985). Correlation lengths for percolation processes. Ph.D. dissertation, Univ. California.
[17] Nolin, P. (2008). Near-critical percolation in two dimensions. Electron. J. Probab. 13 1562–1623.
[18] van den Berg, J., Járai, A. A. and Vágvölgyi, B. (2007). The size of a pond in 2D invasion percolation. Electron. Comm. Probab. 12 411–420.
[19] van den Berg, J., Peres, Y., Sidoravicius, V. and Vares, M. E. (2008). Random spatial growth with paralyzing obstacles. Ann. Inst. H. Poincaré Probab. Statist. 44 1173–1187.
Mathematical Reviews (MathSciNet): MR2469340
Zentralblatt MATH: 05611479
Digital Object Identifier: doi:10.1214/07-AIHP161
Project Euclid: euclid.aihp/1227287570
[20] Werner, W. (2008). Lectures on two-dimensional critical percolation. Available at arXiv:0710.0856.
[21] Wilkinson, D. and Willemsen, J. F. (1983). Invasion percolation: A new form of percolation theory. J. Phys. A 16 3365–3376.
Mathematical Reviews (MathSciNet): MR725616
Digital Object Identifier: doi:10.1088/0305-4470/16/14/028
[22] Zhang, Y. (1995). The fractal volume of the two-dimensional invasion percolation cluster. Comm. Math. Phys. 167 237–254.
Mathematical Reviews (MathSciNet): MR1316507
Zentralblatt MATH: 0811.60095
Digital Object Identifier: doi:10.1007/BF02100587
Project Euclid: euclid.cmp/1104271992
The Annals of Probability
