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Critical behavior and the limit distribution for long-range oriented percolation. I

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Abstract

We consider oriented percolation on \({\mathbb{Z}}^d\times{\mathbb{Z}}_+\) whose bond-occupation probability is pD( · ), where p is the percolation parameter and D is a probability distribution on \({\mathbb{Z}}^d\) . Suppose that D(x) decays as |x|dα for some α > 0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension \(d_c=2(\alpha\wedge2)\). We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to \(e^{-c|k|^{\alpha\wedge2}}\) for some c > 0.

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References

  1. 1.

    Aizenman M. and Barsky D.J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108: 489–526

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 6.

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

  7. 7.

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

  8. 8.

    Chen L.-C. and Shieh N.-R. (2006). Critical behavior for an oriented percolation with long-range interactions in dimension d > 2. Taiwanese J. Math. 10: 1345–1378

  9. 9.

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

  10. 10.

    Grimmett, G., Hiemer, P.: Directed percolation and random walk. In and Out of Equilibrium. In: Sidoravicius, V. (ed.) Birkhäuser, pp. 273–297 (2002)

  11. 11.

    Hara T. and Slade G. (1990). Mean-field critical behavior for percolation in high dimension. Comm. Math. Phys. 128: 233–391

  12. 12.

    Hara T. and Slade G. (1990). On the upper critical dimension of lattice trees and lattice animals. J. Stat. Phys. 59: 1469–1510

  13. 13.

    Hara T. and Slade G. (1992). Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147: 101–136

  14. 14.

    van der Hofstad R. and Sakai A. (2004). Gaussian scaling for the critical spread-out contact process above the upper critical dimension. Electron. J. Probab. 9: 710–769

  15. 15.

    van der Hofstad R. and Sakai A. (2005). Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions. Probab. Theory Relat. Fields 132: 438–470

  16. 16.

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

  17. 17.

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

  18. 18.

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

  19. 19.

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

  20. 20.

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

  21. 21.

    Russo L. (1981). On the critical percolation probabilities. Z. Warsch. verw. Geb. 56: 229–237

  22. 22.

    Sakai A. (2001). Mean-field critical behavior for the contact process. J. Stat. Phys. 104: 111–143

  23. 23.

    Sakai A. (2007). Lace expansion for the Ising model. Comm. Math. Phys. 272: 283–344

  24. 24.

    Sakai, A.: Diagrammatic bounds on the lace-expansion coefficients for oriented percolation. Unpublished manuscript: arXiv:0708.2897[math.PR] (2007)

  25. 25.

    Slade, G.: The lace expansion and its applications. Lecture Notes in Mathematics, 1879 (2006)

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Correspondence to Akira Sakai.

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Chen, L., Sakai, A. Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Relat. Fields 142, 151–188 (2008). https://doi.org/10.1007/s00440-007-0101-2

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Keywords

  • Critical Exponent
  • Limit Distribution
  • Critical Behavior
  • Contact Process
  • Percolation Parameter