Journal of Statistical Physics

, Volume 55, Issue 5–6, pp 981–995 | Cite as

Scaling inequalities for oriented percolation

  • Richard Durrett
  • Nelson I. Tanaka


We look at seven critical exponents associated with two-dimensional oriented percolation. Scaling theory implies that these quantities satisfy four equalities. We prove five related inequalitites.

Key words

Critical phenomena scaling relations critical exponents correlation lengths renormalized bond construction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. L. Cardy and R. L. Sugar, Directed percolation and Reggeon field theory,J. Phys. A 13:L423–427 (1980).Google Scholar
  2. 2.
    R. Durrett, Oriented percolation in two dimensions,Ann. Prob. 12:999–1040 (1984).Google Scholar
  3. 3.
    R. Durrett and D. Griffeath, Supercritical contact processes onZ. Ann. Prob. 11:1–15 (1983).Google Scholar
  4. 4.
    R. Durrett and R. H. Schonmann, Contact process on a finite set II,Ann. Prob. (1988).Google Scholar
  5. 5.
    R. Durrett, R. H. Schonmann, and N. I. Tanaka, Correlation lengths for oriented percolation,J. Stat. Phys. 55:965–979 (1989).Google Scholar
  6. 6.
    M. E. Fisher, Rigorous inequalities for critical point correlation exponents,Phys. Rev. 180:594–600 (1969).Google Scholar
  7. 7.
    P. Grassberger and A. De La Torre, Reggeon field theory (Schlögl's first model) on a lattice: Monte Carlo calculations of critical behavior,Ann. Phys. (N.Y.)122:373–396 (1979).Google Scholar
  8. 8.
    T. Harris, A lower bound for the critical probability in certain percolation processes,Proc. Camb. Phil. Soc. 56:13–20 (1960).Google Scholar
  9. 9.
    H. Kesten, Scaling relations for 2D-percolation,Commun. Math. Phys. 109(1):109–156 (1987).Google Scholar
  10. 10.
    B. G. Nguyen, Correlation length and its critical exponents for percolation processes,J. Stat. Phys. 46:517–523 (1987).Google Scholar
  11. 11.
    H. Tasaki, Hyperscaling inequalities for percolation,Commun. Math. Phys. 113:49–65 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Richard Durrett
    • 1
  • Nelson I. Tanaka
    • 1
    • 2
  1. 1.Cornell UniversityIthaca
  2. 2.University of Sao PauloSao PauloBrazil

Personalised recommendations