Abstract
We review the relations of percolation and related geometrical models to equilibrium statistical mechanical models. These relations are of two sorts: 1) Percolation and related geometrical models have been shown to be equivalent to limits of models in equilibrium statistical mechanics in many cases. 2) Many models exist in which a field subject to thermal disorder is added to the geometrically disordered model so that the two kinds of disorder compete. Throughout this review we attempt to point out places where mathematically exact results might be useful.
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References
S.K. Ma, Modern Theory of Critical Phenomena, Benjamin, Reading, Mass. (1976).
V.K.S. Shante and S. Kirkpatrick, Adv. Physics 20, 325 (1971).
R. Zallen in Fluctuation Phenomena, edited by E.W. Montroll and J.L. Lebowitz, North-Holland, Amsterdam (1979), p. 177.
J.P. Fitzpatrick, R.B. Malt and F. Spaepen, Phys. Lett. A 47, 207 (1974); H. Ottavi, J. Clerc, G. Giraud, J. Roussenq, E. Guyon and C.D. Mitescu, J. Phys. C 11, 1311 (1978). See also reference 3.
R. Zallen, Phys. Rev. B 16, 1426 (1977).
Reference 5. Also, J.W. Halley and W.K. Holcomb, Phys. Rev. Lett. 40, 1670 (1978).
J.W. Halley in Percolation Structures and Processes, ed. by R. Zallen, J. Adler and G. Deutscher, Annals of Israel Phys. Soc. (in press); F. Scholl and K. Binder, Z. Physik B 39, 239 (1980).
H. Muller-Krumbhaar, Phys. Lett. 48A, 459 (1974); A. Coniglio, J. Phys. A 8, 1773 (1975).
H.L. Frisch and J.M. Hammersley, J. Soc. Indust. Appl. Math. 11, 894 (1963); P. Agrawal, S. Redner, P.J. Reynolds and H.E. Stanley, J. Phys. A 12, 2073 (1979).
M. Barma and J.W. Halley, Proc. of the Nucl. Phys. and Sol. St. Symposium, Madras (1979); T. Mai and J.W. Halley in Ordering in Two Dimensions, S. Sinha, ed., Elsevier North Holland (1980), p. 369.
G. Toulouse and P. Pfeuty, Introduction to the Renormalization Group and to Critical Phenomena, Wiley, N.Y. (1977).
Ref. 1: see also Real Space Renormalization, ed. by T.W. Burkhardt and J.M.J. van Leeuwen, vol. 30 of Topics in Current Physics, Springer-Verlag, Berlin (1982).
P.W. Kasteleyn and C.M. Fortuin, J. Phys. Soc. Japan Suppl. 16, 11 (1969); T. Lubensky in La Matière Mal Condensée, ed. by R. Balian, R. Maynard and G. Toulouse, North Holland, Amsterdam (1979), p. 404.
A.B. Harris, T.C. Lubensky, W.K. Holcomb and C. Dasgupta, Phys. Rev. Lett. 35, 327 (1975), M.J. Stephen, Phys. Rev. B 15, 5674 (1977).
D.S. Gaunt, M.F. Sykes and H. Ruskin, J. Phys. A 9, 1899 (1976); R. Fisch and A.B. Harris, Phys. Rev. B 18, 416 (1978).
S. Kirkpatrick, Phys. Rev. Lett. 36, 69 (1976).
P.G. de Gennes, Phys. Lett. A 38, 339 (1972).
See M. Daoud et al., Macromolecules, 8, 804 (1975).
J. des Cloizeaux, J. Phys. (Paris) 36, 281 (1975). For more refined versions of this correspondence, see P.D. Gujrati, Phys. Rev. A 24, 2096 (1981) and J. Phys. A 14 L345 (1981); J.C. Wheeler and P. Pfeuty, Phys. Rev. A 24, 1050 (1981).
P. de Gennes, Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, N.Y. (1979), Chapter 10.
M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974) and reference 1.
M.R. Giri, M.J. Stephen, G.S. Grest, Phys. Rev. B 16, 4971 (1977).
M.J. Stephen, Phys. Lett. A 56, 149 (1976).
Reference 7 and J.W. Halley and M.J. Stephen (unpublished).
M.J. Stephen, Phys. Rev. B 17, 4444 (1978); C. Dasgupta, A.B.Harris and T.C. Lubensky, Phys. Rev. B 17, 1375 (1978).
T.C. Lubensky and J. Isaacson, Phys. Rev. Lett. 41, 829 (1978), 42, 410(E) (1979).
T.C. Lubensky and J. Isaacson, Phys. Rev. A 20, 2130 (1979).
P. Pfeuty and J.C. Wheeler, Phys. Lett. A 84, 493 (1981); F. Rys and W. Helfrich, J. Phys. A 15, 599 (1982); P.D. Gujrati, Phys. Rev. B 27, 4507 (1983).
A. Coniglio and F. Peruggi, J. Phys. A 15, 1873 (1982).
P.J. Scalapino, M. Sears, R.A. Ferrell, Phys. Rev. B 6, 3409 (1972).
A.B. Harris, J. Phys. C 7, 1671 (1974).
See, e.g., Ref. 21.
This is also discussed in Reference 21. Note however, that a sign is wrong in Equation (7.3): \(a = \frac{{4 - n}}{{2(n + 8)}}{\text{ }}\varepsilon {\text{ }} - {\text{ }}\frac{{(n + 2)^2 (n + 28)}}{{4(n + 8)^3 }}{\text{ }}\varepsilon ^2\)
L. Onsager, Phys. Rev. 64, 117 (1944).
Reference 21 and note 33 above.
A.B. Harris and T.C. Lubensky, Phys. Rev. Lett. 33, 1540 (1974).
A. Aharony, Phys. Rev. B 12, 1038 (1974).
G. Grinstein and A. Luther, Phys. Rev. B 13, 1329 (1976).
C. Thompson, Contemporary Physics 19, 203 (1978). Also References 1, 11, 21.
T. Bergstresser, J. Phys. C10, 3831 (1977).
M. Fisher, Phys. Rev. 162, 480 (1967); see also A.B. Harris, J. Phys. C 7, 3082 (1974).
T.C. Lubensky, Phys. Rev. B 15, 311 (1972).
P.G. de Gennes, J. Phys. Lett. 38, 567 (1977).
A.S. Skal and B.I. Shklovskii, Fiz. Tekh. Poluproudn. 8, 1582 (1974) (Sov. Phys. Semicond. 8, 1029 (1975)).
H.E. Stanley, Phys. Rev. 179, 570 (1969).
H.E. Stanley, R.J. Birgeneau, P.J. Reynolds and J. Nicoll, J. Phys. C 9, L553 (1976).
D.J. Wallace and A.P. Young, Phys. Rev. B 17, 2384 (1978).
R.J. Birgeneau, R.A. Cowley, G. Shirane and H.T. Guggenheim, Phys. Rev. Lett. 37, 940 (1976); R.J. Birgeneau, R.A. Cowley, G. Shirane, J.A. Tarvin and H.J. Guggenheim, Phys. Rev. B 21, 317 (1980); R.A. Cowley, R.J. Birgeneau, G. Shirane, H.J. Guggenheim and H. Ikeda, Phys. Rev. B 21, 4038 (1980).
A. Coniglio, Phys. Rev. Lett. 46, 250 (1981); A. Coniglio in Disordered Systems and Localization, ed. by C. Castellani, C. Di Castro and L. Peliti, vol. 149 of Lecture Notes in Physics, Springer-Verlag, Berlin (1981) pp. 51–55.
J.L. van Hemmen and R.G. Palmer, J. Phys. A 15, 3881 (1982); J. Phys. A 12, 563 (1979).
M.J. Stephen and G.S. Grest, Phys. Rev. Lett. 38, 567 (1977).
A bond version of this model is discussed by A.B. Harris, Phys. Rev. Lett. 49, 296 (1982); see also R. Raghavan and D.C. Mattis, Phys. Rev. B 23, 4791 (1981); Y. Shapir, A. Aharony and A.B. Harris, Phys. Rev. Lett. 49, 486 (1982).
A.A. Abrikasov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, N.J. (1963) Chapter 1.
See for example, the papers on this subject in Disordered Systems and Localization, ed. by C. Castellani, C. di Castro and L. Peliti, vol. 149 of Lecture Notes in Physics, Springer-Verlag, Berlin (1981).
H. Kunz and B. Souillard, pp. 213–218, Ref. 54; H. Kunz and B. Souillard, Comm. Math. Phys. 78 201 (1980).
H. Kunz and B. Souillard, J. Physique Lett. 43, L39, 1982.
F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard, R. Weder, Comm. Math. Phys. 88, 387, 1983.
H. Kunz and B. Souillard, J. Physique Lett. 44, L411, 1983.
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Halley, J.W., Dasgupta, C. (1983). Percolation and related systems in equilibrium statistical mechanics. In: Hughes, B.D., Ninham, B.W. (eds) The Mathematics and Physics of Disordered Media: Percolation, Random Walk, Modeling, and Simulation. Lecture Notes in Mathematics, vol 1035. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073263
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DOI: https://doi.org/10.1007/BFb0073263
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