Journal of Statistical Physics

, Volume 78, Issue 3–4, pp 815–825 | Cite as

Ising models, julia sets, and similarity of the maximal entropy measures

  • Yutaka Ishii
Articles

Abstract

We study the phase transition of Ising models on diamondlike hierarchical lattices. Following an idea of Lee and Yang, one can make an analytic continuation of free energy of this model to the complex temperature plane. It is known that the Migdal-Kadanoff renormalization group of this model is a rational endomorphism (denoted byf) of Ĉ and that the singularities of the free energy lie on the Julia setJ(f). The aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onJ(f). Moreover, using this representation, we can show a close relationship between the critical exponent and local similarity of this measure.

Key Words

Ising models diamondlike hierarchical lattices renormalization groups Julia sets maximal entropy measures fractal structure 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T. D. Lee and C. N. Yang, Statistical theory of equations of state and phase transitions, II. Lattice gas and Ising model,Phys. Rev. 87:410–419 (1952).Google Scholar
  2. 2.
    A. A. Migdal, Recurrence equation in gauge field theory.Sov. Phys. JETP 69:810–822, 1457–1467 (1975).Google Scholar
  3. 3.
    L. P. Kadanoff, Notes on Migdal's recursion formulae,Ann. Phys. 100:359–394 (1976).Google Scholar
  4. 4.
    P. M. Bleher and E. Zalis, Asymptotics of the susceptibility for the Ising model on the hierarchical lattices,Commun. Math. Phys. 120:409–436 (1989).Google Scholar
  5. 5.
    P. M. Bleher and M. Yu. Lyubich, Julia sets and complex singularities in hierarchical Ising models.Commun. Math. Phys. 141:453–474 (1991).Google Scholar
  6. 6.
    B. Derrida, L. De Seze, and C. Itzykson, Fractal structure of zeros in hierarchical lattices,J. Stat. Phys. 30:559–570 (1983).Google Scholar
  7. 7.
    B. Derrida, L. De Seze, and J. M. Luck, Oscillatory critical amplitudes in hiearchical models,Commun. Math. Phys. 94:115–127 (1984).Google Scholar
  8. 8.
    B. Derrida, J.-P. Eckmann, and A. Erzan, Renormalization groups with periodic and aperiodic orbits,J. Phys. A: Math. Gen. 16:893–906 (1983).Google Scholar
  9. 9.
    H. Brolin, Invariant sets under iteration of rational functions,Ark. Mat. 6:103–144 (1965).Google Scholar
  10. 10.
    A. Freire, A. Lopes, and R. Mañé, An invariant measure for rational maps.Bol. Soc. Bras. Mat. 14:45–62 (1983).Google Scholar
  11. 11.
    R. Mañé, On the uniqueness of the maximizing measure for rational maps,Bol. Soc. Bras. Mat. 14:27–43 (1983).Google Scholar
  12. 12.
    M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,Ergodic Theory Dynam. Syst. 3:351–385 (1983).Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Yutaka Ishii
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

Personalised recommendations