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Journal of Statistical Physics

, Volume 78, Issue 3–4, pp 841–876 | Cite as

Gaussian limit for critical oriented percolation in high dimensions

  • Bao Gia Nguyen
  • Wei-Shih Yang
Articles

Abstract

In this paper, we consider the spread-out oriented bond percolation models inZ d ×Z withd>4 and the nearest-neighbor oriented bond percolation model in sufficiently high dimensions. Let η n ,n=1, 2, ..., be the random measures defined onR d by
$$\eta _n (A) = \sum\limits_{x \in Z^d } {1_A (x/\sqrt n )1_{\{ (0,0) \to (x,n)\} } } $$
The mean of η n , denoted by\(\bar \eta _n \), is the measure defined by
$$\bar \eta _n (A) = E_p [\eta _n (A)]$$
We use the lace expansion method to show that the sequence of probability measures\([\bar \eta _n (R^d )]^{ - 1} \bar \eta _n \) converges weakly to a Gaussian limit asn→∞ for everyp in the subcritical regime as well as the critical regime of these percolation models. Also we show that for these models the parallel correlation length\(\xi (p)~|p_c - p|^{ - 1} \) asp⇆pc

Key Words

Oreinted percolation connectivity function lace expansion infrared bound Gaussian limit critical exponent parallel correlation length mean-field behavior 

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References

  1. 1.
    M. Aizenman and D. J. Barsky, Sharpness of the phase transition in percolation models,Commun. Math. Phys. 108:489–526 (1987).Google Scholar
  2. 2.
    M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models,J. Stat. Phys. 36:107–143 (1984).Google Scholar
  3. 3.
    D. J. Barsky and M. Aizenman, Percolation critical exponents under the triangle condition.Ann. Prob. (4)19:1520–1536 (1991).Google Scholar
  4. 4.
    C. Bezuidenhout and G. Grimmett, The critical contact process dies out,Ann. Prob. (4)18:1462–1482 (1990).Google Scholar
  5. 5.
    D. Brydges, J. Fröhlich, and A. Sokal, A new proof of the existence and nontriviality of the continuum ϕ24 and ϕ34 quantum field theories,Commun. Math. Phys. 91:141–186 (1983).Google Scholar
  6. 6.
    M. Campanino, J. Chayes, and L. Chayes, Gaussian fluctuations of connectivities in the subcritical regime of percolation.Prob. Theory Related Fields (3)88:269–341 (1991).Google Scholar
  7. 7.
    T. J. Cox and R. Durrett, Oriented percolation in dimensionsd≥4: Bounds and asymptotic formulas,Math. Proc. Camb. Phil. Soc. 93:151–162 (1983).Google Scholar
  8. 8.
    R. Durrett, Some general results concerning the critical exponents of percolation processes,Z. Wahrsch. Verw. Gebiete 69:421–437 (1985).Google Scholar
  9. 9.
    P. Grassberger and A. De La Torre, Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behaviour,Ann. Phys. 122:373–396 (1979).Google Scholar
  10. 10.
    T. Hara, Mean-field critical phenomena for correlation length for percolation in high dimenions,Prob. Theory Related Fields 86:337–385 (1990).Google Scholar
  11. 11.
    T. Hara and G. Slade, Mean-field critical phenomena for percolation in high dimensions,Commun. Math. 128:333–391 (1990).Google Scholar
  12. 12.
    T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals,J. Stat. Phys. 59:1469–1510 (1990).Google Scholar
  13. 13.
    T. Hara and G. Slade, Self-avoiding walk in five or more dimensions. I. The critical behaviour,Commun. Math. Phys. 147:101–136 (1992).Google Scholar
  14. 14.
    T. Hara and G. Slade, The lace expansion for self-avoiding walk in five or more dimensions,Rev. Math. Phys. 4:235–327 (1992).Google Scholar
  15. 15.
    T. Hara and G. Slade, The number and size of branched polymers in high dimensions,J. Stat. Phys. 67:1009–1038 (1992).Google Scholar
  16. 16.
    M. V. Menshikov,Sov. Math. Dokl. 33:856–859 (1986); see also M. V. Menshikov, S. A. Molchanov, and A. F. Sidorenko, Percolation theory and some applications,Itogi Nauki Tekhniki: Teor. Veroyatnost. Matemat. Stat. Teor. Kibernet. 24:53–110 (1980);J. Sov. Math. 42:1766–1810 (1986).Google Scholar
  17. 17.
    B. G. Nguyen and W. S. Yang, Triangle condition for oriented percolation in high dimensions.Ann. Prob. (4)21:1809–1844 (1993).Google Scholar
  18. 18.
    S. P. Obukhov, The problem of directed percolation,Physica 101A:145–155 (1980).Google Scholar
  19. 19.
    G. Slade, The diffusion of self-avoiding random walk in high dimensions,Commun. Math. Phys. 110:661–683 (1987).Google Scholar
  20. 20.
    G. Slade, The scaling limit of self-avoiding random walk in high dimensions,Ann. Prov. 17:91–107 (1989).Google Scholar
  21. 21.
    W. S. Yang and B. G. Nguyen, Gaussian limit for oriented percolation in high dimensions, inProceedings of the Conference on Probability Models in Mathematical Physics (World Scientific, Singapore, 1991), pp. 189–238.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Bao Gia Nguyen
    • 1
  • Wei-Shih Yang
    • 2
  1. 1.Department of MathematicsIllinois Institute of TechnologyChicago
  2. 2.Department of MathematicsTemple University Philadelphia

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