Letters in Mathematical Physics

, Volume 37, Issue 2, pp 137–143 | Cite as

On the extremality of the disordered state for the Ising model on the Bethe lattice

  • Dmitry Ioffe
Article

Abstract

We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k≥2 is extremal if and only if β is less or equal to the spin glass transition value, given by tanh(βcSG= 1/√k.

Mathematics Subject Classifications (1991)

82B20 82B26 82B43 

Key words

Bethe lattice FK representation Ising model 

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References

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    Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M.: Discontinuity of the magnetization in one-dimensional 1/∣x−y2 Ising and Potts models, J. Stat. Phys. 50(1) (1988), 1–40.Google Scholar
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    Bleher, P. M.: Extremity of the disordered phase in the Ising model on the Bethe lattice, Comm. Math. Phys. 128 (1990), 411–419.Google Scholar
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    Bleher, P. M., Ruiz, J. and Zagrebnov, V. A.: On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice, J. Stat. Phys. 79(1/2) (1995), 473–482.Google Scholar
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    Chayes, J. T., Chayes, L., Sethna, J. P. and Thouless, D. J.: A mean field spin glass with short range interactions, Comm. Math. Phys. 106 (1986), 41–89.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Dmitry Ioffe
    • 1
    • 2
  1. 1.Department of MathematicsNorthwestern UniversityU.S.A.
  2. 2.WIASBerlinGermany

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