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(β SGc = 1/√k.
References
Aizenman, M., Chayes, J. T., Chayes, L. and Newman, C. M.: Discontinuity of the magnetization in one-dimensional 1/∣x−y∣2 Ising and Potts models, J. Stat. Phys. 50(1) (1988), 1–40.
Bleher, P. M.: Extremity of the disordered phase in the Ising model on the Bethe lattice, Comm. Math. Phys. 128 (1990), 411–419.
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.
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.
Author information
Authors and Affiliations
Additional information
The work was partially supported by the NSF grant DMS 9504513.
Rights and permissions
About this article
Cite this article
Ioffe, D. On the extremality of the disordered state for the Ising model on the Bethe lattice. Letters in Mathematical Physics 37, 137–143 (1996). https://doi.org/10.1007/BF00416016
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00416016