Journal of Statistical Physics

, Volume 171, Issue 3, pp 462–469 | Cite as

Hyperscaling for Oriented Percolation in \(1+1\) Space–Time Dimensions

  • Akira Sakai


Consider nearest-neighbor oriented percolation in \(d+1\) space–time dimensions. Let \(\rho ,\eta ,\nu \) be the critical exponents for the survival probability up to time t, the expected number of vertices at time t connected from the space–time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality \(d\nu \ge \eta +2\rho \), which holds for all \(d\ge 1\) and is a strict inequality above the upper-critical dimension 4, becomes an equality for \(d=1\), i.e., \(\nu =\eta +2\rho \), provided existence of at least two among \(\rho ,\eta ,\nu \). The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. [6].


Oriented percolation Critical behavior Critical exponent Hyperscaling Box-crossing property 



This work was initiated when I started preparation for the Summer School in Mathematical Physics, held at the University of Tokyo from August 25 through 27, 2017. I am grateful to the organizers, Yasuyuki Kawahigashi and Yoshiko Ogata, for the opportunity to speak at the summer school and meet with many researchers in the laminar-turbulent flow transition. Finally, I would like to thank Alessandro Giuliani for his support during the refereeing process and anonymous referees for valuable comments to the earlier version to this paper.


  1. 1.
    Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36, 107–143 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, L.-C., Sakai, A.: Critical behavior and the limit distribution for long-range oriented percolation. I. Probab. Theory Relat. Fields 142, 151–188 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, L.-C., Sakai, A.: Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation. Ann. Prob. 39, 507–548 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343, 725–745 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Duminil-Copin, H., Tassion, V., Teixeira, A.: The box-crossing property for critical two-dimensional oriented percolation. Probab. Theory Relat. Fields (to appear). arXiv:1610.10018
  7. 7.
    Grimmett, G.: Percolation, 2nd edn. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Grimmett, G., Hiemer, P.: Directed percolation and random walk. In and Out of Equilibrium. Sidoravicius, V., Birkhäuser (eds.) 273–297 (2002)Google Scholar
  9. 9.
    van der Hofstad, R., Holmes, M.: The survival probability and \(r\)-point functions in high dimensions. Ann. Math. 178, 665–685 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    van der Hofstad, R., Slade, G.: A generalised inductive approach to the lace expansion. Probab. Theory Relat. Fields 122, 389–430 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kesten, H.: Scaling relations for \(2D\)-percolation. Commun. Math. Phys. 109, 109–156 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Menshikov, M.V.: Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33, 856–859 (1986)zbMATHGoogle Scholar
  13. 13.
    Nguyen, B.G., Yang, W.-S.: Triangle condition for oriented percolation in high dimensions. Ann. Prob. 21, 1809–1844 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nguyen, B.G., Yang, W.-S.: Gaussian limit for critical oriented percolation in high dimensions. J. Stat. Phys. 78, 841–876 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ódor, G.: Universality classes in nonequilibrium lattice systems. Rev. Mod. Phys. 76, 663–724 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheor. verw. Geb 43, 39–48 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. verw. Geb. 56, 229–237 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sakai, A.: Mean-field critical behavior for the contact process. J. Stat. Phys. 104, 111–143 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sakai, A.: Hyperscaling inequalities for the contact process and oriented percolation. J. Stat. Phys. 106, 201–211 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sano, M., Tamai, K.: A universal transition to turbulence in channel flow. Nat. Phys. 12, 249–253 (2016)CrossRefGoogle Scholar
  21. 21.
    Seymour, P.D., Welsh, D.J.A.: Percolation probabilities on the square lattice. Ann. Discret. Math. 3, 227–245 (1978)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations