Skip to main content
SpringerLink
Log in

On the renormalization group transformation for scalar hierarchical models

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

Abstract

We give a new proof for the existence of a non-Gaussian hierarchical renormalization group fixed point, using what could be called a beta-function for this problem. We also discuss the asymptotic behavior of this fixed point, and the connection between the hierarchical models of Dyson and Gallavotti.

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. Dyson, F. J.: Existence of a phase transition in a one-dimensional Ising ferromagenet. Commun. Math. Phys.12, 91 (1969)

    Google Scholar 

  2. Baker, G. A.: Ising, model with a scaling interaction. Phys. Rev. B5, 2622 (1972)

    Google Scholar 

  3. Bleher, P. M., Sinai, Ya. G.: Critical indices for Dyson's asymptotically hierarchical models. Commun. Math. Phys.45, 347 (1975)

    Google Scholar 

  4. Collet, P., Eckmann, J.-P.: A renormalization group analysis of the hierarchical model in statistical mechanics. Lecture Notes in Physics, Vol. 74. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  5. Gallavotti, G.: Some aspects of the renormalization problems in statistical mechanics. Memorie dell' Accademia dei Lincei15, 23 (1978)

    Google Scholar 

  6. Koch, H., Wittwer, P.: A Non-Gaussian renormalization group fixed point for hierarchical scalar lattice field theories. Commun. Math. Phys.106, 495 (1986)

    Google Scholar 

  7. Koch, H., Wittwer, P.: Computing bounds on critical indices. In: Non-Linear Evolution and Chaotic Phenomena. Gallavotti, G., Zweifel, P. (eds.). NATO ASI Series B: Phys. Vol. 176. New York: Plenum Press 1988

    Google Scholar 

  8. Balaban, T.: Ultraviolet stability in field theory. The Φ 43 -model. In: Scaling and Self-Similarity in Physics, Fröhlich, J. (ed.). Boston: Birkhäuser 1983

    Google Scholar 

  9. Gawędzki, K., Kupiainen, A.: Rigorous renormalization group and asymptotic freedom. In: Scaling and Self-Similarity in Physics, Fröhlich, J. (ed.). Boston: Birkhäuser 1983

    Google Scholar 

  10. See e. g. Theorem (18.2a) in: Marden, M.: Geometry of polynomials. Mathematical Surveys, Number 3. Providence, RI: American Mathematical Society 1966

    Google Scholar 

  11. See e. g. Theorems (4.1.8) and (4.2.1) in: Boas, R. P.: Entire Functions. New York: Academic Press 1954

    Google Scholar 

  12. See e. g. Theorem (5.1) in Hirsch, M. W., Pugh, C. C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  13. See. e. g. Theorem (IX.9) in: Reed, M., Simon, B.: Methods of modern mathematical physics, I: Functional analysis. New York: Academic Press 1980

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, 78712, Austin, TX, USA

    Hans Koch

  2. Département de Physique Théorique, Université de Genève, CH-1211, Genève 4, Switzerland

    Peter Wittwer

Authors
  1. Hans Koch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Wittwer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

Supported in Part by the National Science Foundation under Grant No. DMS-8802590

Supported in Part by the Swiss National Science Foundation

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koch, H., Wittwer, P. On the renormalization group transformation for scalar hierarchical models. Commun.Math. Phys. 138, 537–568 (1991). https://doi.org/10.1007/BF02102041

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02102041

Keywords

Search

Navigation