Journal of Statistical Physics

, Volume 54, Issue 1–2, pp 163–170 | Cite as

On the upper critical dimensions of random spin systems

  • Hal Tasaki


A set of critical exponent inequalities is proved for a large class of classical random spin systems. The inequalities imply rigorous (and probably the optimal) lower bounds for the upper critical dimensions, i.e.,du≥4 for regular and random ferromagnets,du≥6 for spin glasses and random field systems.

Key words

Random spin systems critical exponent inequalities upper critical dimensions 


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  1. 1.
    H. Tasaki,Commun. Math. Phys. 113:49 (1987).Google Scholar
  2. 2.
    D. S. Fisher,Phys. Rev. Lett. 54:1063 (1985).Google Scholar
  3. 3.
    H. Nishimori,Prog. Theor. Phys. 66:1169 (1981); M. Schwartz and A. Soffer, Phys. Rev. Lett.55:2499 (1985); J. T. Chayes, L. Chayes, D. S. Fisher, and T. Spencer,Phys. Rev. Lett. 57:2999 (1987).Google Scholar
  4. 4.
    M. E. Fisher, inCritical Phenomena, M. S. Green, ed. (Academic Press, New York, 1972).Google Scholar
  5. 5.
    K. G. Wilson and J. Kogut,Phys. Rep. 12C:75 (1974).Google Scholar
  6. 6.
    M. E. Fisher,Phys. Rev. 180:594 (1969).Google Scholar
  7. 7.
    L. L. Liu, R. I. Joseph, and H. E. Stanley,Phys. Rev. B 6:1963 (1972).Google Scholar
  8. 8.
    J. T. Chayes, L. Chayes, and J. Fröhlich,Commun. Math. Phys. 100:399 (1985); M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman,J. Phys. A 20:L313 (1987).Google Scholar
  9. 9.
    A. B. Harris,J. Phys. C 7:1671 (1974).Google Scholar
  10. 10.
    K. Binder and A. P. Young,Rev. Mod. Phys. 58:801 (1986); D. S. Fisher and D. A. Huse, Equilibrium behavior of the spin-glass ordered phase, preprint.Google Scholar
  11. 11.
    S. F. Edwards and P. W. Anderson,J. Phys. F 5:965 (1975); D. Sherington and S. Kirkpatrick,Phys. Rev. Lett. 35:1792 (1975); G. Parisi,Phys. Rev. Lett. 43:1754 (1979); J. T. Chayes, L. Chayes, J. P. Sethna, and D. J. Thouless,Commun. Math. Phys. 106:41 (1986).Google Scholar
  12. 12.
    Y. Imry and S. K. Ma,Phys. Rev. Lett. 35:1399 (1975); D. S. Fisher, J. Fröhlich, and T. Spencer,J. Stat. Phys. 34:863 (1984); J. Imbrie,Commun. Math. Phys. 98:145 (1985); J. Bricmont and A. Kupiainen,Phys. Rev. Lett. 59:1829 (1987);Commun. Math. Phys., to appear.Google Scholar
  13. 13.
    G. Grinstein,Phys. Rev. Lett. 37:944 (1976); A. Aharony, Y. Imry, and S. K. Ma,Phys. Rev. Lett. 37:1364 (1976); G. Parisi and N. Sourlas,Phys. Rev. Lett. 43:744 (1979); M. Schwartz and A. Soffer,Phys. Rev. B 33:2059 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Hal Tasaki
    • 1
  1. 1.Physics DepartmentPrinceton UniversityPrinceton

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