Journal of Statistical Physics

, Volume 88, Issue 1–2, pp 231–268 | Cite as

Griffiths' singularities in diluted ising models on the Cayley tree

  • J. C. A. Barata
  • D. H. U. Marchetti


The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.

Key Words

Lee-Yang singularities Griffiths' singularities infinite differentiability metastable states 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AW] M. Aizenman and J. Wehr, “Rounding Effects of Quenched Randomness on First-Order Phase Transitions,”Commun. Math. Phys. 130:489–528 (1990).MATHMathSciNetCrossRefADSGoogle Scholar
  2. [BG-R] F. A. Bosco and S. Goulart Rosa jr, “Fractal Dimension of the Julia Set Associated with the Yang-Lee Zeros of the Ising Model on the Cayley Tree,”Europhys. Lett. 4:1103–1108 (1987).ADSGoogle Scholar
  3. [DKP] H. von Dreifus, A. Klein, and J. F. Perez, “Taming Griffiths Singularities: Infinite Differentiability of the Correlations,”Commun. Math. Phys. 170:21–39 (1995).MATHCrossRefADSGoogle Scholar
  4. [Fi] M. E. Fisher, “The Theory of Condensation and the Critical Point,”Physics 3:255–283 (1967).Google Scholar
  5. [F] J. Fröhlich, “Mathematical Aspects of the Physics of Disordered Systems,” inCritical Phenomena, Random System and Gauge Theories, K. Osterwalder and R. Stora, eds. (Elsevier, Amsterdam, 1986).Google Scholar
  6. [G] R. B. Griffiths, “Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet,”Phys. Rev. Lett. 23:17–19 (1969).CrossRefADSGoogle Scholar
  7. [KG] P. J. Kortman and R. B. Griffiths, “Density of Zeros on the Lee-Yang Circle for Two Ising Ferromagnets,”Phys. Rev. Lett. 27:1439–1442 (1971).CrossRefADSGoogle Scholar
  8. [LY] 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).MATHMathSciNetCrossRefADSGoogle Scholar
  9. [MW] B. McCoy and T. T. Wu, “The Two Dimensional Ising Model” (Cambridge, Harvard University Press, 1973).Google Scholar
  10. [Mo] J. L. Monroe, “Comment on “Fractal Dimension of the Julia Set Associated with the Yang-Lee Zeros of the Ising Model on the Cayley Tree,” F. A. Boscoet al.,”Europhysics Letters 29:187–188 (1995).Google Scholar
  11. [S] A. Sütő, “Weak Singularity and Absence of Metastability in Random Ising Ferromagnets,”J. Phys. A: Math. Gen. 15:L749-L752 (1982).CrossRefGoogle Scholar
  12. [T] E. C. Titchmarsh, “The Theory of Functions” (Oxford University Press, Second Edition, 1939).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • J. C. A. Barata
    • 1
  • D. H. U. Marchetti
    • 1
  1. 1.Instituto de Física, Universidade de São PauloSão PauloBrazil

Personalised recommendations