On the Aizenman exponent in critical percolation
The probabilities of clusters spanning a hypercube of dimension two to seven along one axis of a percolation system under criticality were investigated numerically. We used a modified Hoshen-Kopelman algorithm combined with Grassberger’s “go with the winner” strategy for the site percolation. We carried out a finite-size analysis of the data and found that the probabilities confirm Aizenman’s proposal of the multiplicity exponent for dimensions three to five. A crossover to the mean-field behavior around the upper critical dimension is also discussed.
PACS numbers64.60.Ak 64.60.Fr
Unable to display preview. Download preview PDF.
- 5.L. N. Shchur, in Computer Simulation Studies in Condensed-Matter Physics XII, Ed. by D. P. Landau, S. P. Lewis, and H.-B. Schüttler (Springer-Verlag, Heidelberg, 2000); cond-mat/9906013.Google Scholar
- 7.D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1992).Google Scholar
- 8.A. Bunde and S. Havlin, in Fractals and Disordered Systems, Ed. by A. Bunde and S. Havlin (Springer-Verlag, Berlin, 1996, 2nd ed.).Google Scholar
- 15.G. Paul, R. M. Ziff, and H. E. Stanley, Phys. Rev. E 64, 026115 (2001); cond-mat/0101136.Google Scholar
- 16.P. Grassberger, cond-mat/0202144.Google Scholar
- 19.G. Andronico, A. Coniglio, and S. Fortunato, hep-lat/0208009.Google Scholar