Abstract
We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.
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Communicated by A. Jaffe
Research supported by the NSF under Grant No. DMR-8314625
Research supported by the DOE under Grant No. DE-AC02-83ER13044
Research supported by the NSF under Grant No. DMR-8503544
Research supported by the NSF under Grant No. DMR-8319301
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Chayes, J.T., Chayes, L., Sethna, J.P. et al. A mean field spin glass with short-range interactions. Commun.Math. Phys. 106, 41–89 (1986). https://doi.org/10.1007/BF01210926
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DOI: https://doi.org/10.1007/BF01210926