Contemporary Mathematics Volume 437, 2007 Mean-Field Spin Glass Models from the Cavity-ROSt Perspective Michael Aizenman, Robert Sims, and Shannon L. Starr ABSTRACT. The Sherrington-Kirkpatrick spin glass model has been studied as a source of insight into the statistical mechanics of systems with highly di- versified collections of competing low energy states. The goal of this summary is to present some of the ideas which have emerged in the mathematical study of its free energy. In particular, we highlight the perspective of the cavity dy- namics, and the related variational principle. These are expressed in terms of Random Overlap Structures (ROSt), which are used to describe the possible states of the reservoir in the cavity step. The Parisi solution is presented as reflecting the ansatz that it suffices to restrict the variation to hierarchal struc- tures which are discussed here in some detail. While the Parisi solution was proven correct, through recent works of F. Guerra and M. Talagrand, the rea- sons for the effectiveness of the Parisi ansatz still remain to be elucidated. We question whether this could be related to the quasi-stationarity of the special subclass of ROSts, given by Ruelle's hierarchal 'random probability cascades' (also known as GREM). 0. An outline The Sherrington-Kirkpatrick spin glass model has been studied as a source of insight into statistical mechanics of systems with highly diversified collections of patterns for the minimization of the free energy, or energy. The model is based on a Hamiltonian which incorporates interactions with high levels of frustration and disorder. The goal of this article is to present some of the ideas which have emerged in the study of the SK model, and in particular highlight an approach for the analysis of its free energy influenced by the cavity perspective. The discussion is organized as follows. In Section 1 we present the Sherrington-Kirkpatrick model [22], and comment on some of its basic features and puzzles. A more general version of the model is presented in Appendix C. Among the essential features exhibited by these models is the presence of rich diversity of low energy configurations. A proposal for a solution of the SK model was developed in a series of works, driven by the astounding insight of G. Parisi [17]. An essential feature of the proposed solution is the ansatz that at low temperatures the model's Gibbs states exhibit a hierarchal structure. The Parisi approach was further clarified by Mezard, Parisi and Virasoro [15], and @2007 American Mathematical Society 1 http://dx.doi.org/10.1090/conm/437/08422
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