Skip to main content

The Incipient Infinite Cluster for High-Dimensional Unoriented Percolation

Abstract

We consider bond percolation on \(\mathbb{Z}^d\) at the critical occupation density p c for d>6 in two different models. The first is the nearest-neighbor model in dimension d≫6. The second model is a “spread-out” model having long range parameterized by L in dimension d>6. In the spread-out case, we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as |x|→∞ when d>6 and L is sufficiently large. We also give a general criterion for this convergence to hold, which is satisfied in the case d≫6 in the nearest-neighbor model by work of Hara.(12) We further give a second construction, by taking p<p c , defining a measure \(\mathbb{Q}^p \) and taking its limit as pp c . The limiting object is the high-dimensional analogue of Kesten's incipient infinite cluster (IIC) in d=2. We also investigate properties of the IIC such as bounds on the growth rate of the cluster that show its four-dimensional nature. The proofs of both the existence and of the claimed properties of the IIC use the lace expansion. Finally, we give heuristics connecting the incipient infinite cluster to invasion percolation, and use this connection to support the well-known conjecture that for d>6 the probability for invasion percolation to reach a site x is asymptotic to c|x|−(d−4) as |x|→∞.

This is a preview of subscription content, access via your institution.

References

  1. A. Aharony, Y. Gefen, and A. Kapitulnik, Scaling at the percolation threshold above six dimensions, J. Phys. A. 17:197(1984).

    Google Scholar 

  2. M. Aizenman, On the number of incipient spanning clusters, Nuclear Phys. B 485:551–582 (1997).

    Google Scholar 

  3. S. Alexander, G. Grest, H. Nakanishi, and T. Witten Jr., Branched polymer approach to the structure of lattice animals and percolation clusters, J. Phys. A. 17:185(1984).

    Google Scholar 

  4. M. T. Barlow, Random walks on supercritical percolation clusters, Preprint (2003).

  5. M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36:107–143 (1984).

    Google Scholar 

  6. I. Benjamini, H. Kesten, Y. Peres, and O. Schramm, Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12,... to appear in Ann. Math.

  7. C. E. Bezuidenhout and G. R. Grimmett, The critical contact process dies out, Ann. Probab. 18:1462–1482 (1990).

    Google Scholar 

  8. C. Borgs, J. T. Chayes, H. Kesten, and J. Spencer, The birth of the infinite cluster cluster: finite-size scaling in percolation, Commun. Math. Phys. 224:153–204 (2001).

    Google Scholar 

  9. D. C. Brydges and T. Spencer, Self-avoiding walk in 5 or more dimensions, Commun. Math. Phys. 97:125–148 (1985).

    Google Scholar 

  10. J. T. Chayes, L. Chayes, and C. M. Newman, Stochastic geometry of invasion percolation, Commun. Math. Phys. 101:383–407 (1985).

    Google Scholar 

  11. G. Grimmett, Percolation, 2nd Ed. (Springer, Berlin, 1999).

    Google Scholar 

  12. T. Hara, Critical two-point functions for nearest-neighbor high-dimensional self-avoiding walk and percolation, In preparation.

  13. T. Hara, R. van der Hofstad, and G. Slade, Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models, Ann. Probab. 31:349–408 (2003).

    Google Scholar 

  14. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128:333–391 (1990).

    Google Scholar 

  15. T. Hara and G. Slade, The self-avoiding-walk and percolation critical points in high dimensions, Comb. Probab. Comput. 4:197–215 (1995).

    Google Scholar 

  16. T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents, J. Statist. Phys. 99:1075–1168 (2000).

    Google Scholar 

  17. R. van der Hofstad, F. den Hollander, and G. Slade, Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions, Commun. Math. Phys. 231:435–461 (2002).

    Google Scholar 

  18. R. van der Hofstad and G. Slade, Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions, Ann. Inst. H. Poincaré Probab. Statist. 39:413–485 (2003).

    Google Scholar 

  19. A. A. Járai, Invasion percolation and the incipient infinite cluster in 2D, Commun. Math. Phys. 236:311–334 (2003).

    Google Scholar 

  20. A. A. Járai, Incipient infinite percolation clusters in 2D, Ann. Probab. 31:444–485 (2003).

    Google Scholar 

  21. H. Kesten, The critical probability of bond percolation on the square lattice equals \({\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\), Commun. Math. Phys. 74:41–59 (1980).

    Google Scholar 

  22. H. Kesten, Percolation Theory for Mathematicians (Birkhäuser, Boston, 1982).

    Google Scholar 

  23. H. Kesten, The incipient infinite cluster in two-dimensional percolation, Probab. Theory Related Fields, 73:369–394 (1986).

    Google Scholar 

  24. H. Kesten, Subdiffusive behavior of random walk on a random cluster, Ann. Inst. Henri Poincaré 22:425–487 (1986).

    Google Scholar 

  25. G. F. Lawler, O. Schramm, and W. Werner, One-arm exponent for critical 2D percolation, Electron. J. Probab. 7, 13pp., electronic (2002).

    Google Scholar 

  26. N. Madras and G. Slade, The Self-Avoiding Walk (Birkhäuser, Boston, 1993).

    Google Scholar 

  27. J. Neveu, Mathematical Foundations of the Calculus of Probability (Holden-Day, San Francisco, 1965).

    Google Scholar 

  28. C. M. Newman and D. L. Stein, Ground-state structure in a highly disordered spin-glass model, J. Statist. Phys. 82:1113–1132 (1996).

    Google Scholar 

  29. B. G. Nguyen and W.-S. Yang, Triangle condition for oriented percolation in high dimensions, Ann. Probab. 21:1809–1844 (1993).

    Google Scholar 

  30. B. G. Nguyen and W.-S. Yang, Gaussian limit for critical oriented percolation in high dimensions, J. Statist. Phys. 78:841–876 (1995).

    Google Scholar 

  31. R. Rammal and G. Toulouse, Random walk on fractal structures and percolation clusters, J. Phys. Lett. 44:L13–L22 (1983).

    Google Scholar 

  32. L. Russo, On the critical percolation probabilities, Z. Wahrsch. Verw. Gebiete 56:229–237 (1981).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

van der Hofstad, R., Járai, A.A. The Incipient Infinite Cluster for High-Dimensional Unoriented Percolation. Journal of Statistical Physics 114, 625–663 (2004). https://doi.org/10.1023/B:JOSS.0000012505.39213.6a

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSS.0000012505.39213.6a

  • percolation
  • lace expansion
  • critical phenomena
  • incipient infinite cluster