A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics pp 19-37 | Cite as

# The existence of a non-trivial fixed point

Part I. Heuristics

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## 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

- [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

- [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 ϕ_{ɛ} is a C^{N} function of ɛ ⩾ 0, for all N and ɛ sufficiently small, so that the ɛ-expansion for the critical indices, and more knowledge about ϕ_{ɛ} follows. The proof of the existence of ϕ_{ɛ} 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.

- [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

- [19]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

- [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

- [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

- [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

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

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