Spin Glass Models

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The proposed research concerns spin glass models and addresses important questions related to the Sherringon-Kirkpatrick and the generalized Sherrington-Kirkpatrick models that include characterization of replica symmetric region and phase transitions, replica symmetry breaking, uniqueness of Parsi measure and its properties, strong Ghirlanda-Guerra identities, chaos and ultrametricity conjectures, boundedness of the operator norm of the covariance matrix and the limit of its eigenvalue distribution function in the high temperature ragion, and the TAP equations. The proposal also includes a question of computing the free energy in the diluted spin glass model.The rigorous theory of spin glasses will have broad scientific impact on a number of areas and disciplines, including physics, condensed matter theory, probability and analysis. For example, understanding the ultrametric picture could eventually lead to understanding of the famous Parsi trick which is widely used in physics. Success will probably help in making progress in other spin glass models and more physical systems of great interest such as quantum spin glasses.