Communications in Mathematical Physics

, Volume 125, Issue 1, pp 43–69 | Cite as

The large-scale limit of Dyson's hierarchical vector-valued model at low temperatures. the marginal case\(c = \sqrt 2\)

  • P. M. Bleher
  • P. Major


In this paper we construct the equilibrium states of Dyson's vector-valued hierarchical model with parameter\(c = \sqrt 2\) at low temperatures and describe their large-scale limit. The analogous problems for\(\sqrt 2\) <c<2 and 1<c<\(\sqrt 2\) were solved in our papers [1] and [2]. In the present case the large-scale limit is similar to the case\(\sqrt 2\) <c<2, i.e. it is a Gaussian self-similar field with long-range dependence in the direction orthogonal to and a field consisting of independent Gaussian random variables in the direction parallel with the magnetization. The main difference between the two cases is that now the normalizing factor in the direction of the magnetization contains, beside the square-root of the volume, a logarithmic term too.


Neural Network Statistical Physic Equilibrium State Complex System Nonlinear Dynamics 
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  1. 1.
    Bleher, P.M., Major, P.: Renormalization of Dyson's hierarchical vector-valuedΦ 4-model at low temperatures. Commun. Math. Phys.95, 487–532 (1984)Google Scholar
  2. 2.
    Bleher, P.M., Major, P.: The large-scale limit of Dyson's hierarchical vector-valued model at low temperatures. The non-Gaussian case. Ann. Inst. Henri Poincaré. Phys. Théor.49 (1988)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • P. M. Bleher
    • 1
  • P. Major
    • 2
  1. 1.Keldysh Institute for Applied Mathematics of the Academy of Sciences of the USSRMoscow A-47USSR
  2. 2.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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