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Generalized contour models

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Abstract

The paper is devoted to a survey and systematic exposition of the technique of obtaining cluster expansions for lattice Gibbs fields in the low-temperature region in the case of a finite or countable number of basic states.

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Literature cited

  1. V. M. Gertsik, “Conditions for the uniqueness of a Gibbs state for lattice models with a finite potential of interaction,” Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 2, 448–462 (1976).

    Google Scholar 

  2. J. Glimm, A. Jaffe, and T. Spencer, “Phase transitions in ϕ2/4 models of quantum field theory,” in: Euclidean Quantum Field Theory. The Markov Approach [in Russian], Moscow (1978), pp. 46–131.

  3. R. L. Dobrushin, “The existence of a phase transition in the two-dimensional and three-dimensional Ising models,” Teor. Veroyatn. Primen.,10, No. 2, 209–230 (1965).

    Google Scholar 

  4. V. A. Malyshev, “Soliton sectors in lattice models with continuous time,” Funks. Anal. Prilozhen.,13, No. 1, 31–41 (1979).

    Google Scholar 

  5. V. A. Malyshev, “Cluster expansions in lattice models of statistical physics and quantum field theory,” Usp. Mat. Nauk,35, No. 2, 3–53 (1980).

    Google Scholar 

  6. V. A. Malyshev, “Perturbations of Gibbs fields,” in: Multicomponent Random Systems [in Russian], Nauka, Moscow (1978), pp. 258–276.

    Google Scholar 

  7. V. A. Malyshev and R. A. Minlos, “The spectrum of the transfer matrix of a Gibbs field at low temperatures,” in: Texts of Reports, III International Vilnius Conference on Probability Theory [in Russian], Vilnius, Vol. 2 (1981), pp. 21–22.

    Google Scholar 

  8. V. A. Malyshev and E. N. Petrova, “The duality transformation for Gibbs random fields,” in: Probability Theory. Mathematical Statistics. Theory of Cybernetics [in Russian], Vol. 18 (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1981), pp. 3–51.

    Google Scholar 

  9. V. A. Malyshev and E. N. Petrova, “Generalized contour models,” in: Texts of Reports, III International Vilnius Conference on Probability Theory, Vilnius, Vol. 2 (1981), pp. 23–24.

    Google Scholar 

  10. R. A. Minlos and Ya. G. Sinai, “The phenomenon of “phase separation” at low temperature in some lattice models of a gas. I,” Mat. Sb.,73, No. 3, 375–448 (1967).

    Google Scholar 

  11. R. A. Minlos and Ya. G. Sinai, “The phenomenon of ‘phase separation’ at low temperatures in some lattice models of a gas, II,” Tr. Mosk. Mat. Obshch.,19, 113–178 (1968).

    Google Scholar 

  12. E. N. Petrova, “Low-temperature expansions in the Z model,” Tr. Coord. Conference on the Theory of Multicomponent Random Systems, Tyumen (1980).

  13. S. A. Pirogov and Ya. G. Sinai, “Phase transitions of the first kind for small perturbations of the Ising model,” Funkts. Anal. Prilozhen.,8, No. 1, 25–31 (1974).

    Google Scholar 

  14. Ya. G. Sinai, The Theory of Phase Transitions. Rigorous Results [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  15. Yu. A. Terletskii, “Gibbs random fields with continuous values in the low-temperature region,” in: Interacting Markov Processes and Their Application in Biology [in Russian], Pushchino (1979), pp. 49–64.

  16. Yu. A. Terletskii, “Gibbs random fields in the low-temperature region,” Dokl. Akad. Nauk SSSR,246, No. 3, 540–544 (1979).

    Google Scholar 

  17. T. Balaban and K. Gawendzki, “A low temperature expansion for the pseudoscalar Yukawa model of quantum fields in two space-time dimensions,” Preprint (1980).

  18. Henk Beijeren and Garret S. van Sylvester, “Phase transitions for continuous-spin Ising ferromagnets,” J. Funk. Anal.,28, No. 2, 145–167 (1978).

    Google Scholar 

  19. A. B. Bortz and R. B. Griffiths, “Phase transitions in anisotropic classical Heisenberg ferromagnets,” Commun. Math. Phys.,26, No. 2, 102–108 (1972).

    Google Scholar 

  20. J. Bricmont, J. L. Lebowitz, and C. E. Pfister, “Low-temperature expansion for continuous-spin Ising models,” Commun. Math. Phys.,78, No. 1, 117–135 (1980).

    Google Scholar 

  21. D. C. Brydges, “A rigorous approach to Debye screening in dilute classical Coulomb systems,” Commun. Math. Phys.,58, 313–350 (1978).

    Google Scholar 

  22. D. C. Brydges and P. Feberbush, “Debye screening in classical statistical mechanics,” Mathematical Problems in Theoretical Physics. Proc. Int. Conf. Lausanne, 20–25 Aug., 1979, Lect. Notes Phys.,116, 151–155 (1980).

    Google Scholar 

  23. D. C. Brydges and P. Federbush, “Debye screening,” Commun. Math. Phys.,73, 197–246 (1980).

    Google Scholar 

  24. J. Fröhlich, R. Israel, E. H. Lieb, and B. Simon, “Phase transitions and reflection positivity. I. General theory and long range lattice models,” Commun. Math. Phys.,62, No. 1, 1–34 (1978).

    Google Scholar 

  25. J. Fröhlich, R. Israel, E. H. Lieb, and B. Simon, “Phase transitions and reflection positivity. II. Lattice systems with short-range and Coulomb interactions,” J. Statist. Phys.,22, No. 3, 297–347 (1980).

    Google Scholar 

  26. J. Fröhlich and E. H. Lieb, “Phase transitions in anisotropic lattice spin systems,” Commun. Math. Phys.,60, No. 3, 233–267 (1978).

    Google Scholar 

  27. K. Gawedzky, “Existence of three phases for a P(g4)2 model of quantum fields,” Commun. Math. Phys.,59, 117–142 (1978).

    Google Scholar 

  28. B. Gidas, “The Glimm-Jaffe -Spencer expansion for the classical boundary conditions and coexistence of phases in the λϕ 442 Euclidean (quantum) field theory,” Ann. Phys. (USA),118, No. 1, 18–83 (1979).

    Google Scholar 

  29. R. B. Griffiths, “Peierls proof of spontaneous magnetization in a two-dimensional Ising ferromagnet,” Phys. Rev.,136A, 437–439 (1964).

    Google Scholar 

  30. W. Holsztynski and J. Slawny, “Phase transitions in ferromagnetic spin systems at low temperatures,” Commun. Math. Phys.,66, No. 2, 147–166 (1979).

    Google Scholar 

  31. J. Z. Imbrie, “Mass spectrum of the two-dimensional λϕ4−1/4 ϕ2μϕ quantum field theory,” Commun. Math. Phys.,78, No. 2, 169–200 (1980).

    Google Scholar 

  32. J. Z. Imbrie, “Cluster expansions and mass spectrum for P(ϕ)2 models possessing many phases,” Harvard Univ. Thesis (1980).

  33. J. A. Imbrie, “Phase diagrams and cluster expansions for low temperature P(ϕ)2 models. I. Phase diagrams. II. The Schwinger functions,” Harvard Univ. Preprint (1981).

  34. K. Kuroda, “Phase transitions in classical Heisenberg models,” Tsukuba J. Math.,2, 135–143 (1978).

    Google Scholar 

  35. C. C. F. J. Lage, “Systems de spin vectoriel: existence d'une transition de phase pour le model d'-Heisenberg antiferromagnetique et anisotrope,” These Docteur en Physique, Marseille (1980).

  36. V. A. Malyshev, “Phase transitions in classical Heisenberg ferromagnet model with arbitrary parameter of anisotropy,” Commun. Math. Phys.,40, No. 1, 75–82 (1975).

    Google Scholar 

  37. R. E. Peierls, “On Ising's model of ferromagnetism,” Proc. Cambridge Phil. Soc.,32, Part 3, 477–481 (1936).

    Google Scholar 

  38. T. Spencer, “The mass gap for the P(ϕ)2 quantum field model with a strong external field,” Commun. Math. Phys.,39, 63–76 (1974).

    Google Scholar 

  39. S. Summers, “The phase diagram for a two-dimensional Bose quantum field model,” Harvard Univ. Thesis (1979).

  40. S. Summers, “On the phase diagram of a P(ϕ)2 quantum field model,” Ann. Inst. Henri Poincare,34, 173–229 (1981).

    Google Scholar 

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Authors
  1. V. A. Malyshev
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  2. R. A. Minlos
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  3. E. N. Petrova
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  4. Yu. A. Terletskii
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Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 19, pp. 3–54, 1982.

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Malyshev, V.A., Minlos, R.A., Petrova, E.N. et al. Generalized contour models. J Math Sci 23, 2501–2533 (1983). https://doi.org/10.1007/BF01084702

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