Skip to main content
Log in

On the universality of crossing probabilities in two-dimensional percolation

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript


Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of widtha and heightb are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratioa/b. In the limits set by the sample size and by the conventions and on the range of the ratioa/b measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. J. Adler, Y. Mair, A. Aharony, and A. B. Harris, Series study of percolation moments in general dimension,Phys. Rev. B 41:9183–9206 (1990).

    Google Scholar 

  2. J. L. Cardy, Finite-size scaling in strips: Antiperiodic boundary conditions,J. Phys. A 17:L961-L964 (1984).

    Google Scholar 

  3. J. L. Cardy, Critical percolation in finite geometries, Preprint (1991).

  4. G. Grimmett,Percolation (Springer, 1989).

  5. H. Kesten,Percolation Theory for Mathematicians (Birkhaüser, 1982).

  6. Rosso, J. F. Gouyet, and B. Sapoval, Determination of percolation probabilities from the use of a concentration gradient,Phys. Rev. 32:6053–6054 (1985).

    Google Scholar 

  7. G. A. F. Seber,Linear Regression Analysis (Wiley, 1977).

  8. D. Stauffer,Introduction to Percolation Theory (Taylor Francis, 1985).

  9. R. M. Ziff, Test of scaling exponents for percolation-cluster experiments,Phys. Rev. Lett. 56:545–548 (1986).

    Google Scholar 

Download references

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

About this article

Cite this article

Langlands, R.P., Pichet, C., Pouliot, P. et al. On the universality of crossing probabilities in two-dimensional percolation. J Stat Phys 67, 553–574 (1992).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:

Key words