Communications in Mathematical Physics

, Volume 138, Issue 3, pp 537–568 | Cite as

On the renormalization group transformation for scalar hierarchical models

  • Hans Koch
  • Peter Wittwer


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.


Neural Network Statistical Physic Complex System Asymptotic Behavior Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dyson, F. J.: Existence of a phase transition in a one-dimensional Ising ferromagenet. Commun. Math. Phys.12, 91 (1969)Google Scholar
  2. 2.
    Baker, G. A.: Ising, model with a scaling interaction. Phys. Rev. B5, 2622 (1972)Google Scholar
  3. 3.
    Bleher, P. M., Sinai, Ya. G.: Critical indices for Dyson's asymptotically hierarchical models. Commun. Math. Phys.45, 347 (1975)Google Scholar
  4. 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 1978Google Scholar
  5. 5.
    Gallavotti, G.: Some aspects of the renormalization problems in statistical mechanics. Memorie dell' Accademia dei Lincei15, 23 (1978)Google Scholar
  6. 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. 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 1988Google Scholar
  8. 8.
    Balaban, T.: Ultraviolet stability in field theory. The Φ34-model. In: Scaling and Self-Similarity in Physics, Fröhlich, J. (ed.). Boston: Birkhäuser 1983Google Scholar
  9. 9.
    Gawędzki, K., Kupiainen, A.: Rigorous renormalization group and asymptotic freedom. In: Scaling and Self-Similarity in Physics, Fröhlich, J. (ed.). Boston: Birkhäuser 1983Google Scholar
  10. 10.
    See e. g. Theorem (18.2a) in: Marden, M.: Geometry of polynomials. Mathematical Surveys, Number 3. Providence, RI: American Mathematical Society 1966Google Scholar
  11. 11.
    See e. g. Theorems (4.1.8) and (4.2.1) in: Boas, R. P.: Entire Functions. New York: Academic Press 1954Google Scholar
  12. 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 1977Google Scholar
  13. 13.
    See. e. g. Theorem (IX.9) in: Reed, M., Simon, B.: Methods of modern mathematical physics, I: Functional analysis. New York: Academic Press 1980Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Hans Koch
    • 1
  • Peter Wittwer
    • 2
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Département de Physique ThéoriqueUniversité de GenèveGenève 4Switzerland

Personalised recommendations