The existence of a non-trivial fixed point
Part of the Lecture Notes in Physics book series (LNP, volume 74)
Part I. Heuristics
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KeywordsPerturbation Theory Hierarchical Model Implicit Function Theorem Hermite Polynomial Critical Index
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The breakthrough in the computation of a non-trivial fixed point was the paper
Theorem 3.1 is taken from the paper
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  known from constructive field theory.
The reference  is
- 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
- 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
- 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
- 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
- M. CASSANDRO, G. JONA-LASINIO: Asymptotic behaviour of the auto-covariance function and violation of strong mixing (Preprint).Google Scholar
© Springer-Verlag 1978