Some general results concerning the critical exponents of percolation processes

  • Richard Durrett


In this paper we will give some results concerning the critical exponents of percolation processes which are valid for “any” model. These results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on one side for what happens in general. These results and their proofs are closely related to their analogues for the Ising model.


Stochastic Process Probability Theory General Result Mathematical Biology Ising Model 
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  1. Aizenman, M., Newman, C.: Tree graph inequalities and critical behavior in percolation models. J. Statist Phys. [To appear 1984]Google Scholar
  2. Athreya, K., Ney, P.: Branching Processes. Berlin Heidelberg New York: Springer 1972Google Scholar
  3. Bondesson, L.: Angående problem 3129 eller något om spel på enarmad bandit. Elementa. 62, 129–133 (1979)Google Scholar
  4. Durrett, R., Griffeath, D.: Supercritical contact processes on Z. Ann. Probab. 11, 1–15 (1983)Google Scholar
  5. Durrett, R.: An introduction to oriented percolation. Ann. Probab. 12, 999–1040 (1984)Google Scholar
  6. Dwass, M.: The total progeny in a branching process and a related random walk. J. Appl. Probab. 6, 682–686 (1969)Google Scholar
  7. Feller, W.: An Introduction to Probability Theory and Its Applications, Vol.I, third edition. New York: Wiley 1968Google Scholar
  8. Griffeath, D.: Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724. Berlin-Heidelberg-New York: Springer 1979Google Scholar
  9. Griffeath, D.: The basic contact processes. Stochastic Processes Appl. 11, 151–185 (1981)Google Scholar
  10. Jagers, P.: Branching Processes with Biological Applications. New York: Wiley 1975Google Scholar
  11. Kesten, H.: Analyticity properties and power law estimates of functions in percolation theory. J. Statist. Phys. 25, 717–756 (1981)Google Scholar
  12. Kesten, H.: Percolation Theory for Mathematicians. Boston: Birkhauser 1982Google Scholar
  13. Simon, B.: Correlation inequalities and the decay of correlations in ferromagnets. Comm. Math. Phys. 77, 111–126 (1980)Google Scholar
  14. Smythe, R., Wierman, J.: First passage percolation on the square lattice. Lecture Notes Mathematics 671. New York-Heidelberg-Berlin: Springer 1978Google Scholar
  15. Sokal, A.: More inequalities for critical exponents. J. Stat. Phys. 25, 25–50 (1981)Google Scholar
  16. Spencer, J., editor: Paul Erdös: The Art of Counting. Cambridge, Mass.: MIT Press 1973Google Scholar
  17. van den Berg, J., Kesten, H.: Inequalities with application to percolation and reliability. J. Appl. Prob., to appearGoogle Scholar
  18. Durrett, R., B. Nguyen, B.: (1985) Thermodynamic inequalities for percolation [Submitted to Comm. Math. Phys. 98, to appear]Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Richard Durrett
    • 1
  1. 1.Dept. of MathematicsUniversity of CaliforniaLos AngelesUSA

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