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The existence of a non-trivial fixed point

Part I. Heuristics
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Part of the Lecture Notes in Physics book series (LNP, volume 74)

Keywords

Perturbation Theory Hierarchical Model Implicit Function Theorem Hermite Polynomial Critical Index 
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References

The breakthrough in the computation of a non-trivial fixed point was the paper

  1. [16]
    P.M. BLEHER, Ja.G. SINAI: Critical Indices for Dyson's Asymptotically Hierarchical Models, Commun. Math. Phys. 45 347. (1975).CrossRefGoogle Scholar

Theorem 3.1 is taken from the paper

  1. [17]
    M.G. CRANDALL, P.H. RABINOWITZ: Bifurcation from Simple Eigen-values. J. Funct. Anal. 8, 321 (1971).CrossRefGoogle Scholar

The proof of Bleher and Sinai used the “method of the separatrix”. Improving slightly on their method, we showed in that ϕɛ is a CN function of ɛ ⩾ 0, for all N and ɛ sufficiently small, so that the ɛ-expansion for the critical indices, and more knowledge about ϕɛ follows. The proof of the existence of ϕɛ we give in these Lectures Notes is new and has not appeared before. It relies on hypercontractive estimates, cf [36] known from constructive field theory.

  1. [18]
    P. COLLET, J.-P. ECKMANN. The ε-Expansion for the Hierarchical Model. Commun. Math. Phys. 55, 67 (1967).CrossRefGoogle Scholar

The reference [19] is

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    I.M. GELFAND, G.E. SSCHILOW: Verallgemeinerte Funktionen (Distributionen) Band II, Berlin, 1962, VEB Deutscher Verlag der Wissenschaften.Google Scholar

The results of the numerical calculations of Bleher can be found in

  1. [20]
    P.M. BLEHER: Critical indices for models with long range forces (Numerical Calculations). Preprint. Inst. of Applied Math., Acad. Sci. SSSR (1975).Google Scholar

The case √2 < c < 2 has been discussed in great detail in

  1. [21]
    P.M. BLEHER: A second order phase transition in some ferromagnetic models. Trudy Mosc. Math. Obshestvo33, 155 (1975).Google Scholar

The results on the critical indices have been summarized in

  1. [22]
    P.M. BLEHER, Ja.G. SINAI: Critical indices for systems with slowly decaying interaction. Zh.Eksp. Teor. Fiz. 67 391 (1974) [Sov. Phys. JETP. 40, 195 (1975)].Google Scholar

Theorem 3.8. is a variant of an argument suggested by Nappi-Hegerfeldt and given in

  1. [23]
    M. CASSANDRO, G. JONA-LASINIO: Asymptotic behaviour of the auto-covariance function and violation of strong mixing (Preprint).Google Scholar
  2. [24]
    G.C. HEGERFELDT: Prime field decompositions and infinitely divisible states on Borcher's tensor algebra. Commun. math. Phys. 45, 137 (1975).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1978

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