Advertisement

Communications in Mathematical Physics

, Volume 128, Issue 2, pp 333–391 | Cite as

Mean-field critical behaviour for percolation in high dimensions

  • Takashi Hara
  • Gordon Slade
Article

Abstract

The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv2 in (i) and (ii), wherev2 is the critical exponent for the correlation length.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics High Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aizenman, M.: Geometric analysis of φ4 fields and Ising models, Parts I and II. Commun. Math. Phys.86, 1–48 (1982)CrossRefGoogle Scholar
  2. 2.
    Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys.108, 489–526 (1987)CrossRefGoogle Scholar
  3. 3.
    Aizenman, M., Fernández, R.: On the critical behaviour of the magnetization in high dimensional Ising models. J. Stat. Phys.44, 393–454 (1986)CrossRefGoogle Scholar
  4. 4.
    Aizenman, M., Graham, R.: On the renormalized coupling constant and the susceptibility in λφd4 field theory and the Ising model in four dimensions. Nucl. Phys. B225 [FS9], 261–288 (1983)CrossRefGoogle Scholar
  5. 5.
    Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Commun. Math. Phys.111, 505–531 (1987)CrossRefGoogle Scholar
  6. 6.
    Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behaviour in percolation models. J. Stat. Phys.36, 107–143 (1984)CrossRefGoogle Scholar
  7. 7.
    Aizenman, M., Simon, B.: Local Ward identities and the decay of correlations in ferromagnets. Commun. Math. Phys.77, 137–143 (1980)CrossRefGoogle Scholar
  8. 8.
    Barsky, D.J., Aizenman, M.: Percolation critical exponents under the triangle condition. Preprint (1988)Google Scholar
  9. 9.
    van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability, J. Appl. Prob.22, 556–569 (1985)Google Scholar
  10. 10.
    Broadbent, S.R., Hammersley, J.M.: Percolation processes. I. Crystals and mazes. Proc. Camb. Phil. Soc.53, 629–641 (1957)Google Scholar
  11. 11.
    Brydges, D.C., Fröhlich, J., Sokal, A.D.: A new proof of the existence and nontriviality of the continuum φ24 and φ34 quantum field theories. Commun. Math. Phys.91, 141–186 (1983)CrossRefGoogle Scholar
  12. 12.
    Brydges, D.C., Spencer, T.: Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys.97, 125–148 (1985)CrossRefGoogle Scholar
  13. 13.
    Chayes, J.T., Chayes, L.: On the upper critical dimension of Bernoulli percolation. Commun. Math. Phys.113, 27–48 (1987)CrossRefGoogle Scholar
  14. 14.
    Essam, J.W.: Percolation Theory. Rep. Prog. Phys.43, 833–912 (1980)CrossRefGoogle Scholar
  15. 15.
    Fröhlich, J.: On the triviality of φd4 theories and the approach to the critical point in\(d\mathop > \limits_{( = )} 4\) dimensions. Nucl. Phys. B200 [FS4], 281–296 (1982)CrossRefGoogle Scholar
  16. 16.
    Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions, and continuous symmetry breaking. Commun. Math. Phys.50, 79–95 (1976)CrossRefGoogle Scholar
  17. 17.
    Grimmett, G.: Percolation, Berlin Heidelberg New York: Springer 1989Google Scholar
  18. 18.
    Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.28, 790–795 (1957)Google Scholar
  19. 19.
    Hammersley, J.M.: Bornes supérieures de la probabilité critique dans un processus de filtration. In: Le Calcul des Probabilités et ses Applications 17–37 CNRS Paris (1959)Google Scholar
  20. 20.
    Hara, T.: Mean field critical behaviour of correlation length for percolation in high dimensions. Preprint (1989)Google Scholar
  21. 21.
    Hara, T., Slade, G.: On the upper critical dimension of lattice trees and lattice animals. Submitted to J. Stat. Phys.Google Scholar
  22. 22.
    Hara, T., Slade, G.: UnpublishedGoogle Scholar
  23. 23.
    Kac, M., Uhlenbeck, G.E., Hemmer, P.C.: On the van der Waals theory of the vapor-liquid equilibrium. I. Discussion of a one-dimensional model. J. Math. Phys.4, 216–288 (1963)CrossRefGoogle Scholar
  24. 24.
    Kesten, H.: Percolation theory and first passage percolation. Ann. Probab.15, 1231–1271 (1987)Google Scholar
  25. 25.
    Lawler, G.: The infinite self-avoiding walk in high dimensions. To appear in Ann. Probab. (1989)Google Scholar
  26. 26.
    Lebowitz, J.L., Penrose, O.: Rigorous treatment of the van der Waals-Maxwell theory of the liquid-vapor transition. J. Math. Phys.7, 98–113 (1966)CrossRefGoogle Scholar
  27. 27.
    Menshikov, M.V., Molchanov, S.A., Sidorenko, A.F.: Percolation theory and some applications, Itogi Nauki i Tekhniki (Series of Probability Theory, Mathematical Statistics, Theoretical Cybernetics)24, 53–110 (1986). English translation. J. Soviet Math.42, 1766–1810 (1988)Google Scholar
  28. 28.
    Nguyen, B.G.: Gap exponents for percolation processes with triangle condition. J. Stat. Phys.49, 235–243 (1987)CrossRefGoogle Scholar
  29. 29.
    Park, Y.M.: Direct estimates on intersection probabilities of random walks. To appear in J. Stat. Phys.Google Scholar
  30. 30.
    Russo, L.: On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete.56, 229–237 (1981)CrossRefGoogle Scholar
  31. 31.
    Slade, G.: The diffusion of self-avoiding random walk in high dimensions. Commun. Math. Phys.110, 661–683 (1987)CrossRefGoogle Scholar
  32. 32.
    Slade, G.: The scaling limit of self-avoiding random walk in high dimensions. Ann. Probab.17, 91–107 (1989)Google Scholar
  33. 33.
    Slade, G.: Convergence of self-avoiding random walk to Brownian motion in high dimensions. J. Phys. A: Math. Gen.21, L417-L420 (1988)CrossRefGoogle Scholar
  34. 34.
    Slade, G.: The lace expansion and the upper critical dimension for percolation, Lectures notes from the A.M.S. Summer Seminar, Blacksburg, June 1989Google Scholar
  35. 35.
    Sokal, A.D.: A rigorous inequality for the specific heat of an Ising or φ4 ferromagnet. Phys. Lett.71A, 451–453 (1979)Google Scholar
  36. 36.
    Sokal, A.D., Thomas, L.E.: Exponential convergence to equilibrium for a class of random walk models. J. Stat. Phys.54, 797–828 (1989)CrossRefGoogle Scholar
  37. 37.
    Stauffer, D.: Introduction to percolation theory. Taylor and Francis, London Philadelphia (1985)Google Scholar
  38. 38.
    Tasaki, H.: Hyperscaling inequalities for percolation. Commun. Math. Phys.113, 49–65 (1987)CrossRefGoogle Scholar
  39. 39.
    Tasaki, H.: Private communicationGoogle Scholar
  40. 40.
    Yang, W., Klein, D.: A note on the critical dimension for weakly self avoiding walks. Prob. Th. Rel. Fields79, 99–114 (1988)CrossRefGoogle Scholar
  41. 41.
    Ziff, R.M., Stell, G.: Critical behaviour in three-dimensional percolation: Is the percolation threshold a Lifshitz point? Preprint (1988)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Takashi Hara
    • 1
  • Gordon Slade
    • 2
  1. 1.Courant Institute of Mathematical SciencesN.Y.U.New YorkUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations