Inequalities for γ and Related Critical Exponents in Short and Long Range Percolation

  • C. M. Newman
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 8)


We relate the cluster size distribution Pn(p) at the percolation critical point, p = p|1c|0, to the critical exponent γ (ΣnPn(p) ≈ |pc — p| as p ↑ pc). If P(pc) > 0 (i.e., if P(p) is discontinuous at p = pc), then γ ≥ 2. If Pn(pc) ≈ n-1–1/δ as n → ∞, then γ ≥ 2(1 – 1/δ). Related inequalities are yalid for γr (ΣnrPn(p) ≈ |pc — p| r P as p ↑ pc) and γ r ' (defined analogously as p ↓ pc) when r > 1/δ: γr, γ r ' ≥ 2(r — 1/δ). These results are yalid for Bernoulli site or bond percolation on d-dimensional lattices for any d with p the site or bond occupation probability. They are also valid for long range translation invariant Bernoulli bond percolation with p the occupation probability for bonds of some given length.


Critical Exponent Percolation Theory Percolation Model Cluster Size Distribution Bond Percolation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AB]
    Aizenman, M. and Barsky, D. J., Proof of the sharpness of the phase transition in translation invariant percolation models, Rutgers University preprint, in preparation.Google Scholar
  2. [ACCIN]
    Aizenman, M., Chayes, J. T., Chayes, L., Imbrie, J. and Newman, C. M., An intermediate phase with slow decay of correlations in one-dimensional 1/|x — y|2 percolation, Ising and Potts models, in preparation.Google Scholar
  3. [AKN]
    Aizenman, M., Kesten, H. and Newman, C. M., Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation, Rutgers University/Cornell University/University of Arizona preprint, in preparation.Google Scholar
  4. [AN1]
    Aizenman, M. and Newman, C. M., Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36 (1984), 107–143.MathSciNetMATHCrossRefGoogle Scholar
  5. [AN2]
    Aizenman, M. and Newman, C. M., Discontinuity of the percolation density in one-dimensional 1/|x – y|2 percolation models, Rutgers University/University of Arizona preprint, 1986.Google Scholar
  6. [BK]
    van den Berg, J. and Keane, M., On the continuity of the percolation probability function, Contemporary Mathematics 26 (1984), 61–65.MATHGoogle Scholar
  7. [CC]
    Chayes, J. T. and Chayes, L., An inequality for the infinite cluster density in Bernoulli percolation, Phys. Rev. Lett. 56 (1986), 1619–1622.MathSciNetCrossRefGoogle Scholar
  8. [CCN]
    Chayes, J. T., Chayes, L. and Newman, C. M., Bernoulli percolation above threshold: an invasion percolation analysis, Ann. Prob., to appear.Google Scholar
  9. [G]
    Grimmett, G. R., On the differentiability of the number of clusters per vertex in the percolation model, J. London Math. Soc. (2) 23 (1981), 372–384.MathSciNetMATHCrossRefGoogle Scholar
  10. [H]
    Hammersley, J. M., Percolation processes. Lower bounds for the critical probability, Ann Math. Statist. 28 (1957), 790–795.MathSciNetMATHCrossRefGoogle Scholar
  11. [K1]
    Kesten, H., The critical probability of bond percolation on the square lattice equals 1/2, Comm. Math. Phys. 74 (1980), 41–59.MathSciNetMATHCrossRefGoogle Scholar
  12. [K2]
    Kesten, H., Percolation Theory for Mathematicians, Birkhauser, 1982.MATHGoogle Scholar
  13. [K3]
    Kesten, H., A scaling relation at criticality for 2D-percolation, in Proceedings of the IMA workshop on percolation theory and ergodic theory of infinite particle systems (H. Kesten, Ed.), IMA Volumes in Mathematics and its Applications, Springer-Verlag, to appear.Google Scholar
  14. [K4]
    Kesten, H., Scaling relations for 2D-percolation, Institute for Mathematics and its Applications (Minneapolis) preprint, 1986.Google Scholar
  15. [KS]
    Klein, S. T. and Shamir, E., An algorithmic method for studying percolation clusters, Stanford Univ. Dept. of Computer Science, Report No. STAN-CS-82–933 (1982).Google Scholar
  16. [N1]
    Newman, C. M. Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperature, University of Arizona preprint (1981), published as an appendix to Percolation theory: a selective survey of rigorous results, to appear in Proceedings of the SIAM Workshop on Multiphase Flow (G. Papanicolaou, Ed.).Google Scholar
  17. [N2]
    Newman, C. M., Some critical exponent inequalities for percolation, J. Stat. Phys., to appear.Google Scholar
  18. [NS]
    Newman, C. M. and Schulman, L. S., One-dimensional 1/|j – i|s percolation models: the existence of a transition for s ≤ 2, Comm. Math. Phys. 104 (1986), 547–571.MathSciNetMATHCrossRefGoogle Scholar
  19. [PS]
    Pike, R. and Stanley, H. E., Order propagation near the percolation threshold, J. Phys. A 14 (1981), L169–L177.CrossRefGoogle Scholar
  20. [R]
    Russo, L., On the critical percolation probabilities, Z. Wahrsch. verw. Geb. 56 (1981), 229–237.MATHCrossRefGoogle Scholar
  21. [S]
    Schulman, L. S., Long range percolation in one dimension, J. Phys. A Lett. 16 (1983), L639–L641.MathSciNetGoogle Scholar
  22. [St]
    Stauffer, D., Scaling properties of percolation clusters, in Disordered Systems and Localization (C. Oastellani, C. DiCastro and L. Peliti, Eds.), Springer, 1981, 9–25.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1987

Authors and Affiliations

  • C. M. Newman
    • 1
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA

Personalised recommendations