Spin glass models from the point of view of spin distributions

Details Spin glass models from the point of view of spin distributions Spin glass models from the point of view of spin distributions

2010-05-16

arxiv.org/abs/1005.2720v1

Mathematics

Probability

Scientific paper

In many spin glass models, due to the symmetry between sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $\sigma: [0,1]^4\to\{-1,+1\}$ and one can think of this function as a generic functional order parameter of the model. In a class of diluted models and in the Sherrington-Kirkpatrick model, we introduce novel perturbations of the Hamiltonians that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington-Kirkpatrick model the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda-Guerra identities and stochastic stability.

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