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Perturbation methods of the theory of Gibbsian fields

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References

  • [Be] S.N. Bernstein, Détermination d’une limite inférieure de la dispersion des sommes de grandeurs liées en chaine singuliére, Math. Sbornik 1 (1936), 29–38.

    MATH  Google Scholar 

  • [Br] D. Brydges, A short course on cluster expansions, Critical Phenomena, Random Systems, Gauge Theory (Les Houches, 1984) (K. Ostewalder and R. Stora, eds.), North-Holland, Amsterdam-New York, 1986, pp. 129–183.

    Google Scholar 

  • [Do56] R.L. Dobrushin, Central Limit Theorem for inhomogeneous Markov chains, Th. Prob. Appl. 1 (1956), 72–88, 365–425.

    MathSciNet  MATH  Google Scholar 

  • [Do65] R.L. Dobrushin, Existence of a phase transition in two and three dimensional Ising models, Th. Prob. Appl. 10 (1965), 193–213.

    Article  MathSciNet  MATH  Google Scholar 

  • [Do70] R.L. Dobrushin, Description of a system of random variables by the help of conditional distributions, Th. Prob. Appl. 15 (1970), 469–497.

    Article  Google Scholar 

  • [Do87] R.L. Dobrushin, Induction on volume and no cluster expansions, Proc. VIII-th Intern. Congress on Math Phys., World scientific, Singapore eds M. Mebkhout and B. Seneor, 1987, pp. 73–91.

    Google Scholar 

  • [Do88] R. Dobrushin, A new approach to the analysis of Gibbs perturbations of Gaussian fields, Selecta Math. Sov. 7 (1988), 221–277.

    MathSciNet  MATH  Google Scholar 

  • [Do94] R. Dobrushin, Estimates of semiinvariants for the Ising model at low temperatures, Preprint 125, ESI, Wien, 1994.

    Google Scholar 

  • [DM] R. Dobrushin and M. Martirosjan, Non-finite perturbation of Gibbsian fields, Theor. and Math. Phys. 74 (1988), 10–20.

    Article  Google Scholar 

  • [DN] R. Dobrushin and B.S. Nahapetian, Strong convexity of the pressure for lattice systems of classical statistical physics, Theor. and Math. Phys. 20 (1974), 782–790.

    Article  MathSciNet  Google Scholar 

  • [DN] R. Dobrushin and S. Shlosman, Consructive criterion for the uniqueness of Gibbs fields, Stat. Phys. and Dynamical Systems. Rigorous Results, Progr. in Phys., 10 20 (1974), 782–790.

    Google Scholar 

  • [DW] R. Dobrushin and V. Warstat, Completely analytical interactions with infinite values, Probabl. Th. Rel. Fields 84 (1990), 335–359.

    Article  MathSciNet  MATH  Google Scholar 

  • [Du] R.M. Dudley, Real Analysis and Probability, Wadsworth & Brooks/Cole, Pacific Grove, California, 1989.

    MATH  Google Scholar 

  • [Ge] H.-O. Georgii, Gibbs Measures and Phase Transitions, Walter de Gruyger, Berlin, 1988.

    Book  MATH  Google Scholar 

  • [GJ] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View, Springer-Verlag, New York-Berlin, 1987.

    MATH  Google Scholar 

  • [GJS] J. Glimm, A. Jaffe, and T. Spencer, A convergent expancion about the mean field theory, I, Ann. Phys 101 (1976), 610–630; II, 631–669.

    Article  MathSciNet  Google Scholar 

  • [GK] C. Gruber, H. Kunz, General properties of polymer systems, Comm. Math. Phys. 22 (1971), 133–161.

    Article  MathSciNet  Google Scholar 

  • [Gr] Griffiths R.B., Peierls’ proof of spontaneous magnetization in a two-dimensional Ising ferromagnet, Phys. Rev. 136A (1964), 437–439.

    Article  MathSciNet  MATH  Google Scholar 

  • [IL] I. A. Ibragimov and Y. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noorhoff, Groningen, 1971.

    MATH  Google Scholar 

  • [Ka] L.V. Kantorovich, On transportation of masses, Reports Acad. Sci. USSR 37 (1942), 225–226.

    Google Scholar 

  • [KP] R. Kotecky and D. Preiss, Cluster expansion for abstract polymer models, Comm. Math. Phys. 103 (1986), 491–498.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ma] V.A. Malyshev, Cluster expansions in lattice models of statistical physics and quantum theory of fields, Russian Math. Surveys 35 (1980), no. 2, 1–62.

    Article  MathSciNet  Google Scholar 

  • [MM] V.A. Malyshev and R.A. Minlos, Gibbs Random Fields. Cluster Expansions, Kluver Academic Publishers Group, Dordrecht, 1991.

    Book  MATH  Google Scholar 

  • [Pe] R. Peierls, On Ising’s model of ferromagnetism, Proc. Cambridge Phil. Soc. 32 (1936), 477–481.

    Article  MATH  Google Scholar 

  • [Pre] C.J. Preston, Gibbs states on Countable Sets, Cambridge Univ. Press, London-New York, 1974.

    Book  MATH  Google Scholar 

  • [Pro] J.V. Prokhorov, Convergence of random processes and limit theorems of the theory of probability, Th. Prob. Appl. 1 (1956), 177–238.

    Article  MATH  Google Scholar 

  • [Ra] S.T. Rachev, The Monge-Kantorovich problem on transportation of masses and its applications in stochastics, Th. Prob. Appl. 29 (1984), 625–653.

    MathSciNet  MATH  Google Scholar 

  • [Se] E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics. Lectures Notes in Physics, vol 159, Springer-Verlag, Berlin-New York, 1982.

    Google Scholar 

  • [Sim] B. Simon, The Statistical Mechanics of Lattice Gases, Volume 1, Princeton University Press, Princeton, New Jersey, 1993.

    Book  Google Scholar 

  • [Sin] Ya. G. Sinai, The Theory of Phase Transitions: Rigorous results, Pergamon, London, 1981.

    Google Scholar 

  • [SS] L. Saulis, V.A. Statulevicius, Limit Theorems for Large Deviations, Kluwer Acad. Publ., Dordrecht-Boston-London, 1991.

    Book  MATH  Google Scholar 

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Pierre Bernard

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Dobrushin, R.L. (1996). Perturbation methods of the theory of Gibbsian fields. In: Bernard, P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0095674

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  • DOI: https://doi.org/10.1007/BFb0095674

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