Abstract
Using a simulation technique introduced recently, we study winding clusters in percolation on the torus and the Möbius strip for different aspect ratios. The asynchronous parallelization of the simulation makes very large system and sample sizes possible. Our high accuracy results are fully consistent with predictions from conformal field theory. The numerical results for the Möbius strip and the number distribution of winding clusters on the torus await theoretical explanation. To our knowledge, this study is the first of its kind.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
M. Aizenman, On the number of incipient spanning clusters, Nucl. Phys. B 485:551–582 (1997).
K. Binder and D. Stauffer, Monte Carlo studies of “random” systems, in Applications of the Monte Carlo Method in Statistical Physics, 2nd ed., K. Binder, ed., Topics in Current Physics, Vol. 36 (Springer-Verlag, Berlin/Heidelberg/New York, 1987), pp. 241–275.
J. Cardy, Critical percolation in finite geometries, J. Phys. A: Math. Gen. 25:L201–L206 (1992), preprint hep-th/9111026.
J. Cardy, The number of incipient spanning clusters in two-dimensional percolation, J. Phys. A: Math. Gen. 31:L105–L110 (1998).
P. di Francesco, H. Saleur, and J. B. Zuber, Relations between the coloumb gas picture and conformal invariance of two-dimensional critical models, J. Stat. Phys. 49:57–79 (1987).
J. Hoshen and R. Kopelman, Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm, Phys. Rev. B 14:3438–3445 (1976).
C.-K. Hu and C.-Y. Lin, Universal scaling functions for numbers of percolating clusters on planar lattices, Phys. Rev. Lett. 77:8–11 (1996).
K. Kaneda and Y. Okabe, Finite-size scaling for the Ising model and the Möbius strip and the Klein bottle, Phys. Rev. Lett. 86:2134–2137 (2001).
H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Commun. Math. Phys. 74:41–59 (1980).
R. Langlands, C. Pichet, P. Pouliot, and Y. Saint-Aubin, On the universality of crossing probabilities in two-dimensional percolation, J. Stat. Phys. 67:553–574 (1992).
W. T. Lu and F. Y. Wu, Ising model on nonorientable surfaces: Exact solution for the Möbius strip and the Klein bottle, Phys. Rev. E 63:026107-1-026107-9 (2001).
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, Berlin/Heidelberg/New York, 1966).
M. Matsumoto and T. Nishimura, Dynamic creation of pseudorandom number generators, in Monte Carlo and Quasi-Monte Carlo Methods 1998 (Springer-Verlag, Berlin/Heidelberg/New York, 1998), preprint from http://www.math.h.kyoto-u.ac.jp/∼matumoto/RAND/DC/dc.html.
B. Mendelson, Introduction to Topology (Allyn and Bacon, Inc., Boston/London/ Sydney/Toronto, 1975).
N. R. Moloney and G. Pruessner, Asynchronously parallelized percolation on distributed machines, Phys. Rev. E 67:037701-1–4 (2003), preprint cond-mat/0211240.
H. Nakanishi and H. E. Stanley, Scaling studies of percolation pheonmena in systems of dimensionality two to seven: Cluster numbers, Phys. Rev. B 22:2466–2488 (1980).
M. E. J. Newman and R. M. Ziff, Efficient Monte Carlo algorithm and high-precision results for percolation, Phys. Rev. Lett. 85:4104–4107 (2000).
M. E. J. Newman and R. M. Ziff, Fast Monte Carlo algorithm for site and bond percolation, Phys. Rev. E 64:016706-1–16 (2001).
H. T. Pinson, Critical percolation on the torus, J. Stat. Phys. 75:1167–1177 (1994).
G. Pruessner and N. R. Moloney, Numerical results for crossing, spanning and wrapping in two-dimensional percolation, J. Phys. A: Math. Gen. 36:11213–11228 (2003).
D. C. Rapaport, Cluster number scaling in two-dimensional percolation, J. Phys. A: Math. Gen. 19:219–304 (1986).
R. M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A: Math. Gen. 28:1249–1255 (1995).
R. M. Ziff, Proof of crossing formula for 2d percolation, J. Phys. A: Math. Gen. 28: 6479–6480 (1995).
R. M. Ziff, C. D. Lorenz, and P. Kleban, Shape-dependent universality in percolation, Physica A 266:17–26 (1999).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pruessner, G., Moloney, N.R. Winding Clusters in Percolation on the Torus and the Möbius Strip. Journal of Statistical Physics 115, 839–853 (2004). https://doi.org/10.1023/B:JOSS.0000022376.25660.7b
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000022376.25660.7b