Abstract
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.
Similar content being viewed by others
References
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).
J. L. Cardy, Finite-size scaling in strips: Antiperiodic boundary conditions,J. Phys. A 17:L961-L964 (1984).
J. L. Cardy, Critical percolation in finite geometries, Preprint (1991).
G. Grimmett,Percolation (Springer, 1989).
H. Kesten,Percolation Theory for Mathematicians (Birkhaüser, 1982).
Rosso, J. F. Gouyet, and B. Sapoval, Determination of percolation probabilities from the use of a concentration gradient,Phys. Rev. 32:6053–6054 (1985).
G. A. F. Seber,Linear Regression Analysis (Wiley, 1977).
D. Stauffer,Introduction to Percolation Theory (Taylor Francis, 1985).
R. M. Ziff, Test of scaling exponents for percolation-cluster experiments,Phys. Rev. Lett. 56:545–548 (1986).
Author information
Authors and Affiliations
Rights 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). https://doi.org/10.1007/BF01049720
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01049720