The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions

Artem Sapozhnikov (ETH Zürich)

Abstract


This note is motivated by results of Angel, Goodman, den Hollander and Slade (2008) and Damron, Sapozhnikov and Vagvolgyi (2009) about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on trees and in two dimensions.

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Pages: 775-780

Publication Date: November 30, 2011

DOI: 10.1214/ECP.v16-1684

References

  1. O. Angel, J. Goodman, F. den Hollander and G. Slade. Invasion percolation on regular trees. Ann. Probab. 36 (2008), 420--466. Math. Review 2009b:60290
  2. D.J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition. Ann. Probab. 19 (1991), 1520-1536. Math. Review 93b:60224
  3. J. van den Berg, A. A. Jarai and B. Vagvolgyi. The size of a pond in 2D invasion percolation. Electron. Comm. Probab. 12 (2007), 411--420. Math. Review 2009c:60265
  4. R. Chandler, J. Koplick, K. Lerman and J. F. Willemsen. Capillary displacement and percolation in porous media. J. Fluid Mech. 119 (1982), 249--267.
  5. J. T. Chayes, L. Chayes and C. Newman. The stochastic geometry of invasion percolation. Commun. Math. Phys. 101 (1985), 383--407. Math. Review 87i:82072
  6. M. Damron and A. Sapozhnikov. Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters. Probab. Theor. Rel. Fields. 150(1-2) (2011), 257-294. DOI: 10.1007/s00440-010-0274-y.
  7. M. Damron and A. Sapozhnikov. Limit theorems for $2D$ invasion percolation. To appear in Annals of Probability (2011). arXiv:1005.5696.
  8. M. Damron, A. Sapozhnikov and B. Vagvolgyi. Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37 (2009), 2297--2331. Math. Review 2011a:60343
  9. J. Goodman. Exponential growth of ponds for invasion percolation on regular trees. Preprint (2009). arXiv: 0912:5205.
  10. G. Grimmett. Percolation. (1999) 2nd edition. Springer, Berlin. Math. Review 2001a:60114
  11. O. Haggstrom, Y. Peres and R. Schonmann. Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness. In Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten (M. Bramson and R. Durrett, eds.) (1999) 69--90. Birkhauser, Basel. Math. Review 2000h:60087
  12. T. Hara and G. Slade. Mean-field critical behaviour for percolation in high dimensions. Commun. Math. Phys. 128 (1990), 333-391. Math. Review 91a:82037
  13. A. A. Jarai. Invasion percolation and the incipient infinite cluster in 2D. Commun. Math. Phys. 236 (2003), 311--334. Math. Review 2004c:82111
  14. H. Kesten. The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74(1) (1980), 41-59. Math. Review 82c:60179
  15. H. Kesten. The incipient infinite cluster in two-dimesional percolation. Probab. Theor. Rel. Fields. 73 (1986), 369--394. Math. Review 88c:60196
  16. H. Kesten. Scaling relations for 2D percolation. Commun. Math. Phys. 109 (1987), 109--156. Math. Review 88k:60174
  17. G. F. Lawler, O. Schramm and W. Werner. One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 (2002), 1-13. Math. Review 2002k:60204
  18. R. Lenormand and S. Bories. Description d'un mecanisme de connexion de liaision destine a l'etude du drainage avec piegeage en milieu poreux. C.R. Acad. Sci. 291 (1980), 279--282.
  19. P. Nolin. Near critical percolation in two-dimensions. Electron. J. Probab. 13 (2008), 1562--1623. Math. Review 2009k:60215
  20. S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 (2001), 729-744. Math. Review 2003i:60173
  21. Y. Zhang. The fractal volume of the two-dimensional invasion percolation cluster. Commun. Math. Phys. 167 (1995), 237-254. Math. Review 95k:60255
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