Skip to main content
SpringerLink
Log in

Renormalization of Dyson's hierarchical vector valued φ4 model at low temperatures

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate Dyson's hierarchical vector valued φ4 model at low temperatures. The case\(2 > c > \sqrt 2 \) is considered. The pure phase is constructed, and the existence of its large scale limit is proved. The limit is Gaussian, but an unusual normalization has to be chosen. In the direction of the spontaneous magnetization one has to divide by the square root of the volume, but in the orthogonal direction one has to divide by a different power of the volume for all low temperatures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bateman, G., Erdélyi, A.: Higher transcendental functions, Vol. 2. New York, Toronto, London: McGraw-Hill 1955

    Google Scholar 

  2. Bleher, P.M.: Construction of non-Gaussian self-similar random fields with hierarchical structure. Commun. Math. Phys.84, 557–578 (1982)

    Google Scholar 

  3. Bleher, P.M.: Large deviations results in Dyson's hierarchical model (in Russian). Teor. Verojatnost. Primenen (to appear)

  4. Bleher, P.M., Sinai, Ya. G.: Investigation of the critical point in models of the type of Dyson's hierarchical models. Commun. Math. Phys.33, 23–42 (1973)

    Google Scholar 

  5. Bleher, P.M., Sinai, Ya. G.: Critical indices for Dyson's asymptotically hierarchical models. Commun. Math. Phys.45, 247–278 (1975)

    Google Scholar 

  6. Collet, P., Eckmann, J.P.: A Renormalization group analysis of the hierarchical model in statistical mechanics. In: Lecture Notes in Physics, No. 74. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  7. Sinai, Ya. G.: Theory of phase transitions: rigorous results. London: Pergamon Press 1982

    Google Scholar 

  8. Dyson, F.: An Ising ferromagnet with distribution soflong-range order. Commun. Math. Phys.21, 269–283 (1971)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics of the Soviet Academy of Sciences, Moscow, USSR

    P. M. Bleher

  2. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary

    P. Major

Authors
  1. P. M. Bleher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Major
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Ya. G. Sinai

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bleher, P.M., Major, P. Renormalization of Dyson's hierarchical vector valued φ4 model at low temperatures. Commun.Math. Phys. 95, 487–532 (1984). https://doi.org/10.1007/BF01210834

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01210834

Keywords

Search

Navigation