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High-Dimensional Incipient Infinite Clusters Revisited

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Abstract

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second-named author and Járai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also get estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls.

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References

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Aizenman, M., Barsky, D.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Alexander, S., Orbach, R.: Density of states on fractals: “fractons”. J. Physique (Paris) Lett. 43, L625–L631 (1982)

  5. Barsky, D., Aizenman, M.: Percolation critical exponents under the triangle condition. Ann. Probab. 19, 1520–1536 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. van der Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab. 22, 556–569 (1985)

    Google Scholar 

  7. Borgs, C., Chayes, J., van der Hofstad, R., Slade, G., Spencer, J.: Random subgraphs of finite graphs. II. The lace expansion and the triangle condition. Ann. Probab. 33(5), 1886–1944 (2005)

    Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Chen, L.-C., Sakai, A.: Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Relat. Fields 142, 151–188 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, L.-C., Sakai, A.: Critical behavior and the limit distribution for long-range oriented percolation. II. Spatial correlation. Probab. Theory Relat. Fields 145, 435–458 (2009)

    Google Scholar 

  11. Chen, L.-C., Sakai, A.: Critical two-point functions for long-range statistical-mechanical models in high dimensions. Ann. Probab. (to appear)

  12. Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2006)

    Google Scholar 

  13. Feldman, J., Knörrer, H., Salmhofer, M., Trubowitz, E.: The temperature zero limit. J. Stat. Phys. 94, 113–157 (1999)

    Article  ADS  MATH  Google Scholar 

  14. Feldman, J., Magnen, J., Rivasseau, V., Sénéor, R.: Bounds on completely convergent Euclidean Feynman graphs. Commun. Math. Phys. 98, 273–288 (1985)

    Article  ADS  MATH  Google Scholar 

  15. Grimmett, G.: Percolation, 2nd edn. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  16. Hara, T.: Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36(2), 530–593 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Hara, T., Slade, G.: Mean-field behaviour and the lace expansion. In: Grimmett, G. (ed.) Probability and Phase Transition. Kluwer, Dordrecht (1994)

  19. Hara, T., van der Hofstad, R., Slade, G.: Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31(1), 349–408 (2003)

    Google Scholar 

  20. Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Cambridge Philos. Soc. 56, 13–20 (1960)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. Heydenreich, M., van der Hofstad, R., Hulshof, T.: Random walk on the high-dimensional IIC. Commun. Math. Phys. 1–59 (2014)

  22. Heydenreich, M., van der Hofstad, R., Hulshof, T., Miermont, G.: Backbone scaling limit of the high-dimensional IIC (in preparation)

  23. Heydenreich, M., van der Hofstad, R., Sakai, A.: Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk. J. Stat. Phys. 132(6), 1001–1049 (2008)

    Google Scholar 

  24. van der Hofstad, R., Járai, A: The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114(3–4), 625–663 (2004)

    Google Scholar 

  25. van der Hofstad, R., Slade, G.: A generalised inductive approach to the lace expansion. Probab. Theory Relat. Fields 122(3), 389–430 (2002)

    Google Scholar 

  26. van der Hofstad, R., den Hollander, F., Slade, G.: Construction of the incipient infinite cluster for spread-out oriented percolation above \(4+1\) dimensions. Commun. Math. Phys. 231(3), 435–461 (2002)

    Google Scholar 

  27. van der Hofstad, R., den Hollander, F., Slade, G.: The survival probability for critical spread-out oriented percolation above 4+1 dimensions. II. Expansion. Ann. Inst. H. Poincaré Probab. Stat 5(5), 509–570 (2007)

    Google Scholar 

  28. Janson, S., Marckert, J.-F.: Convergence of discrete snakes. J. Theor. Probab. 18(3), 615–647 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Járai, A.: Incipient infinite percolation clusters in 2D. Ann. Probab. 31(1), 444–485 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kesten, H.: The critical probability of bond percolation on the square lattice equals \(\frac{1}{2}\). Commun. Math. Phys. 74(1), 41–59 (1980)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  31. Kesten, H.: The incipient infinite cluster in two-dimensional percolation. Probab. Theory Relat. Fields 73(3), 369–394 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  32. Kozma, G.: Percolation on a product of two trees. Ann. Probab. 39(5), 1864–1895 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Kozma, G., Nachmias, A.: The Alexander-Orbach conjecture holds in high dimensions. Invent. Math. 178, 635–654 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Kozma, G., Nachmias, A.: Arm exponents in high-dimensional percolation. J. Am. Math. Soc. 24, 375–409 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

  36. Leyvraz, F., Stanley, H.: To what class of fractals does the Alexander-Orbach conjecture apply? Phys. Rev. Lett. 51, 2048–2051 (1983)

    Article  ADS  Google Scholar 

  37. Menshikov, M.: Coincidence of critical points in percolation problems. Soviet Math. Doklady 33, 856–859 (1986)

    Google Scholar 

  38. Schonmann, R.: Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219, 271–322 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  39. Schonmann, R.H.: Mean-field criticality for percolation on planar non-amenable graphs. Commun. Math. Phys. 225(3), 453–463 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. Slade, G.: The lace expansion and its applications, volume 1879 of Lecture Notes in Mathematics. Springer, Berlin (2006). Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, Edited and with a foreword by Jean Picard

  41. Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)

    Google Scholar 

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Acknowledgments

This work is supported by the Netherlands Organization for Scientific Research (NWO). MH and RvdH thank the Institut Henri Poincaré Paris for kind hospitality during their visit in October 2009. We thank the anonymous referees for their useful comments.

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Authors and Affiliations

  1. Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

    Markus Heydenreich

  2. Centrum voor Wiskunde and Informatica, P.O. Box 94079, 1090 GB , Amsterdam, The Netherlands

    Markus Heydenreich

  3. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB , Eindhoven, The Netherlands

    Remco van der Hofstad

  4. Department of Mathematics, The University of British Columbia, Vancouver, BC, Canada

    Tim Hulshof

  5. Pacific Institute for Mathematical Sciences, The University of British Columbia, Vancouver, BC, Canada

    Tim Hulshof

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  1. Markus Heydenreich
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  2. Remco van der Hofstad
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  3. Tim Hulshof
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Correspondence to Tim Hulshof.

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Heydenreich, M., van der Hofstad, R. & Hulshof, T. High-Dimensional Incipient Infinite Clusters Revisited. J Stat Phys 155, 966–1025 (2014). https://doi.org/10.1007/s10955-014-0979-x

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  • DOI: https://doi.org/10.1007/s10955-014-0979-x

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