Abstract
Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 106 sites. We investigate for the square lattice the variable-range percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponentsβ andγ. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponentν. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.
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References
J. Hoshen and R. Kopelman,Phys. Rev. B 14:3438 (1976).
J. M. Hammersley and D. C. Handscomb, inMonte Carlo Methods (Methuen, London, 1964), chap 11; V. S. Shante and S. Kirkpatrick,Adv. Phys. 20:325 (1971); J. W. Essam, inPhase Transitions and Critical Phenomena, C. Domb and M. S. Green, eds. (Academic, New York, 1973); S. Kirkpatrick,Rev. Mod. Phys. 45:574 (1973).
R. Kopelman, E. M. Monberg, F. W. Ochs, and P. N. Prasad,J. Chem. Phys. 62:292 (1975); R. Kopelman, E. M. Monberg, and F. W. Ochs,Chem. Phys. 19:413 (1977).
A. S. Skal, B. I. Shklovski, and A. L. Efros,Sov. Phys.-Solid State 15:961 (1973);Fiz. Tverd. Tela 15:1423 (1973).
G. E. Pike and C. H. Seager,Phys. Rev. B 10:1421, 1435 (1974).
R. Zallen and H. Scher,Phys. Rev. B 4:4471 (1971).
K. Binder,Thin Solid Films 20:367 (1974).
M. E. Fisher, inProc. Intern. School of Physics “Enrico Fermi” on Critical Phenomena, Course 51, M. S. Green, ed. (Academic Press, New York, 1971).
M. E. Fisher and M. N. Barber,Phys. Rev. Lett. 28:1516 (1972).
M. E. Levinshtein, B. I. Shklovskii, M. S. Shur, and A. L. Efros,Sov. Phys.-JETP 42:197 (1976);Zh. Eksp. Teor. Fiz. 69:386 (1975).
A. Sur, J. L. Lebovitz, J. Morr, M. H. Kalos, and S. Kirkpatrick,J. Stat. Phys. 15:345 (1976).
J. Hoshen and R. Kopelman,J. Chem. Phys. 65:2817 (1976);
R. Kopelman, E. M. Monberg, and F. W. Ochs,Chem. Phys. 21:373 (1977)
R. Kopelman,J. Phys. Chem. 80:2191 (1976);
R. Kopelman, inRadiationless Processes in Molecules and Condensed Phases (Vol. 15 of Topics in Applied Physics), F. K. Fong, ed. (Springer, Berlin, 1976);
R. Kopelman, 8th Molecular Crystal Symp., Santa Barbara, California (1977).
A. B. Harris, T. C. Lubensky, W. K. Holcomb, and C. Dasgupta,Phys. Rev. Lett. 35:327(1975).
M. F. Sykes, D. S. Gaunt, and M. Glen,J. Phys. A 9:97 (1976).
J. Roussenq, J. Clerc, G. Giraud, E. Guyon, and H. Ottavi,J. Phys. (Paris) 37:L99 (1976).
L. P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palciauskas, M. Rayl, J. Swift, D. Aspnes, and J. Kane,Rev. Mod. Phys. 39:395 (1967).
J. W. Essam and K. M. Gwilym,J. Phys. C 4:L228 (1971).
A. G. Dunn, J. W. Essam, and J. M. Loveluck,J. Phys. C 8:743 (1975).
M. E. Fisher,Rev. Mod. Phys. 46:597 (1974).
M. F. Sykes, D. S. Gaunt, and M. Glen,J. Phys. A 9:725 (1976).
D. Stauffer,Z. Physik B 25:391 (1976).
M. S. Shur,J. Phys. C 9:L229 (1976).
A. G. Dunn, J. W. Essam, and D. S. Ritchie,J. Phys. C 8:4219 (1975).
M. F. Sykes, D. S. Gaunt, and M. Glen,J. Phys. A 9:1705 (1976).
D. S. Gaunt and M. F. Sykes,J. Phys. A 9:1109 (1976).
J. Hoshen, P. Klymko, and R. Kopelman, unpublished.
D. S. Gaunt, M. F. Sykes, and H. Ruskin,J. Phys. A 9:1899 (1976).
R. Zallen, private communication, October 1977.
R. Kopelman, Excitons in Pure and Mixed Molecular Crystals, inExcited States II, E. C. Lim, ed. (Academic Press, New York, 1975).
H. Scher and R. Zallen,J. Chem. Phys. 53:3759 (1970).
D. Stauffer,Phys. Rev. Lett. 35:394 (1975).
M. F. Sykes, D. S. Gaunt, and M. Glen,J. Phys. A 9:L43 (1976).
J. Hoshen, unpublished.
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 9:715 (1976).
S. Kirkpatrick,Phys. Rev. Lett. 36:69 (1976).
R. Zallen,Phys. Rev. B 16:1426 (1977).
H. L. Frisch, E. Sonnenblick, V. A. Vyssotsky, and J. M. Hammersley,Phys. Rev. 124:1021 (1961).
P. Dean,Proc. Camb. Phil. Soc. 59:397 (1963).
P. Dean and N. F. Bird, National Physical Laboratory, Report No. Ma61, unpublished (1966);Proc. Camb. Phil. Soc. 63:477 (1967).
M. F. Sykes and J. W. Essam,Phys. Rev. 133:A310 (1964).
D. G. Neal,Proc. Camb. Phil. Soc. 71:97 (1972).
B. P. Watson and P. L. Leath,Phys. Rev. B 9:4893 (1974).
P. L. Leath,Phys. Rev. B 14:5046 (1976).
M. F. Sykes, J. L. Martin, and J. W. Essam,J. Phys. A 6:1306 (1973).
M. F. Sykes, M. Glen, and D. S. Gaunt,J. Phys. A 7:L105 (1974).
E. Stoll and T. Schneider,Phys. Rev. A 6:429 (1972).
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Supported by NSF Grant DMR76-07832.
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Hoshen, J., Kopelman, R. & Monberg, E.M. Percolation and cluster distribution. II. layers, variable-range interactions, and exciton cluster model. J Stat Phys 19, 219–242 (1978). https://doi.org/10.1007/BF01011724
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DOI: https://doi.org/10.1007/BF01011724