Abstract
Algorithms for estimating the percolation probabilities and cluster size distribution are given in the framework of a Monte Carlo simulation for disordered lattices for the generalized site-bond problem. The site-bond approach is useful when a percolation process cannot be exclusively described in the context of pure site or pure bond percolation. An extended multiple labeling technique (ECMLT) is introduced for the generalized problem. The ECMLT is applied to the site-bond percolation problem for square and triangular lattices. Numerical data are given for lattices containing up to 16 million sites. An application to polymer gelation is suggested.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. L. Frisch and J. M. Hammersley,J. Soc. Ind. Appl. Math. B 11:894 (1963).
V. K. S. Shante and S. Kirkpatrick,Adv. Phys. 20:325 (1971).
J. W. Essam,Phase Transition and Critical Phenomena, C. Domb and M. S. Green, eds. (Academic Press, New York, 1973), Vol. 2, p. 197.
L. Torelli and A. E. Scheidegger,Pure Appl. Geophys. 89:32 (1971);
P. G. de Gennes and E. Guyon,J. Mecanique 17:1 (1978).
D. Stauffer,J. Chem. Soc. Faraday Trans. II 72:1354 (1976).
R. J. Elliot, B. R. Heap, D. J. Morgan, and G. S. Rushbrooke,Phys. Rev. Lett. 5:366(1960).
G. E. Pike, W. J. Camp, C. H. Seager, and G. L. McVay,Phys. Rev. B 10:4909 (1974).
R. Kikuchi and H. Sato,J. Chem. Phys. 53:2702 (1970).
J. Hoshen and R. Kopelman,J. Chem. Phys. 65:2817 (1976).
R. Kopelman, E. M. Monberg, F. W. Ochs, and P. N. Prasad,Phys. Rev. Lett. 34:1506 (1975).
S. Kirkpatrick,Phys. Rev. Lett. 27:1722 (1971).
C. H. Seager and G. E. Pike,Phys. Rev. B 10:1435 (1974).
M. H. Cohen, I. Webman, and J. Jortner,J. Chem. Phys. 64:2013 (1976).
M. Pollak and I. Reiss,J. Phys. C 9:2339 (1976).
M. F. Sykes, D. S. Gaunt, and J. W. Essam,J. Phys. A 9:L43 (1976).
J. M. Hammersley,Methods Comput Phys. 1:281 (1963).
M. E. Fisher and J. W. Essam,J. Math. Phys. 2:609 (1961).
M. F. Sykes, D. S. Gaunt, and M. Glen,J. Phys. A 9:87, 97, 715, 725 (1976).
H. L. Frisch, S. B. Gordon, V. A. Vyssotsky, and J. M. Hammersley,Bell. Syst. Tech. J. 41:909 (1962).
P. Dean,Proc. Camb. Phil. Soc. 5:397 (1963).
A. S. Skal, B. I. Shklowski, and A. L. Efros,Sov. Phys.-Solid State 15:961 (1973).
G. D. Quinn, G. H. Bishop, and R. Harrison,J. Phys. A 9:L9 (1976).
S. Kirkpatrick,Phys. Rev. Lett. 36:69 (1976).
P. L. Leath,Phys. Rev. Lett. 36:921 (1976).
J. Hoshen and R. Kopelman,Phys. Rev. B 14:3438 (1976).
G. E. Pike and C. H. Seager,Phys. Rev. B 10:1421 (1974).
R. Zallen and H. Scher,Phys. Rev. B 4:4471 (1971).
J. M. Hammersley and D. C. Handscomb, inMonte Carlo Methods (Methuen, London, 1964), p. 135;
K. Binder, ed.,Monte Carlo Methods in Statistical Physics (Springer, Heidelberg, 1979).
J. Hartmanis and J. E. Hopcroft,J. ACM 18:444 (1971).
S. Kirkpatrik,Rev. Mod. Phys. 45:574 (1973).
J. Hoshen, R. Kopelman, and E. M. Monberg,J. Stat. Phys. 19:219 (1978).
H. L. Frisch, J. M. Hammersley, and D. J. A. Welsh,Phys. Rev. 126:949 (1962).
F. Harary,Graph Theory (Addison-Wesley, Reading, Massachusetts, 1969), p. 32.
A. V. Aho, J. E. Hopcroft, and J. D. Ullman,The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, Massachusetts, 1974), p. 129.
B. A. Galler and M. J. Fischer,Comm. ACM 7:301 (1964).
H. Scher and R. Zallen,J. Chem. Phys. 53:3759 (1970).
J. R. B. Whittlesey,Comm. ACM 11:641 (1968).
G. Masaglia, inApplication of Number Theory to Numerical Analysis, S. K. Zaremba, ed. (Academic Press, New York, 1972).
R. C. Tausworthe,Math. Comput. 19:201 (1965).
C. Tanford,Physical Chemistry of Macromolecules (Wiley, New York, 1961), p. 140.
P. J. Flory,J. Am. Chem. Soc. 63:3083, 3091, 3096 (1941).
R. J. Goldberg,J. Am. Chem. Soc. 74:5715 (1952).
C. DeLisi,J. Theor. Biol. 45:555 (1974);
C. DeLisi and A. Perelson,J. Theor. Biol. 62:159 (1976).
H. Ottavi, J. Clerc, G. Giraud, J. Roussenq, E. Guyon, and C. D. Mitescu,J. Phys. C 11:1311 (1978).
H. J. Wintle and P. H. Puhach,J. Stat. Phys. 18:557 (1978).
A. Coniglio, H. E. Stanley, and W. Klein,Phys. Rev. Lett. 42:518 (1979).
P. Agrawal, S. Redner, P. J. Reynolds, and H. E. Stanley, Preprint (1979).
H. Nakanishi and P. J. Reynolds, preprint (1979).
Author information
Authors and Affiliations
Additional information
Supported by NSF Grant DMR76-07832 and NIH Grant NS08116-9.
Rights and permissions
About this article
Cite this article
Hoshen, J., Klymko, P. & Kopelman, R. Percolation and cluster distribution. III. Algorithms for the site-bond problem. J Stat Phys 21, 583–600 (1979). https://doi.org/10.1007/BF01011170
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01011170