Mathematics – Probability
Scientific paper
2011-05-13
Mathematics
Probability
22 pages
Scientific paper
We study the mixed even-spin Sherrington-Kirkpatrick (SK) model with ferromagnetic interaction and parallel many well-known behaviors of the SK model. Our main results are: (i) The thermodynamic limit of the free energy exists and can be represented in terms of the thermodynamic limit of the free energy of the SK model with a variation of external fields. (ii) In the presence of the centered Gaussian external field, the overlap is positive on a dense subset of the inverse temperature parameter. This implies that our model is equivalent to the SK model with a shifted external field in the sense that they possess the same Parisi measure (the limiting law of the overlap) and its Gibbs measures satisfy the extended Ghirlanda-Guerra identities. (iii) We depict a region for the inverse temperature parameter on which the overlap can be controlled quantitatively away from the origin by the magnetization. Our approach is based on the techniques from the study of the Curie-Weiss model and results in convex analysis.
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