Mean-Field Spin Glass Models

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Mean-field spin glass models and, in particular, the Sherrington-Kirkpatrick model were better understood in the past several years following the discovery of the replica symmetry breaking interpolation by Francesco Guerra and the proof of the celebrated Parisi formula for the free energy by Michel Talagrand. The current proposal consists of several directions of research that will attempt to build upon recent progress. One project proposes to study whether the Ghirlanda-Guerra identities for the distribution of the overlaps, which arise from a certain stochastic stability property of the Gibbs measure, imply the Parisi ultrametricity conjecture. Another project concerns a number of natural analogues of the Guerra replica symmetry breaking interpolation for various spin glass models, such as the perceptron, Hopfield, diluted p-spin and p-sat models. In all these models such interpolations formally reproduce the solutions predicted by theoretical physicists, but since the methodology of the proof of the Parisi formula in the Sherrington-Kirkpatrick model does not directly apply to these models, one needs to find new ways to control the error terms in these interpolations. In addition, the proposal includes several other questions regarding the joint distribution of the overlaps in the spherical Sherrington-Kirkpatrick model, properties of the Parisi functional, and characterization of the replica symmetric region in the Sherrington-Kirkpatrick model via the Almeida-Thouless line.Several models in statistical mechanics, called mean-field spin glass models, were originally introduced and studied by theoretical physicists who developed an impressive heuristic theory that gave detailed predictions about the behaviorof these models and that influenced many other areas of research well beyond the scope of the original problems. Rigorous mathematical proofs of some of the physicist's predictions required a number of new ideas and approaches that are likely to be useful in other areas of probability, statistical physics, computer science and statistics. Current proposal will continue research in several promising directions.