Mean-Field Spin Glass Models

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。平均场自旋玻璃模型，特别是谢林顿-柯克帕特里克模型，在过去几年里，随着弗朗西斯科·格拉（Francesco Guerra）发现了复制对称性破切插值和米歇尔·塔拉格兰（Michel Talagrand）证明了著名的巴黎自由能公式，人们对其有了更好的理解。目前的建议包括几个研究方向，这些方向将试图以最近的进展为基础。一个项目提出研究重叠分布的Ghirlanda-Guerra恒等式是否隐含Parisi超对称性猜想，该恒等式源于Gibbs测度的某种随机稳定性。另一个项目涉及许多自然类似于Guerra复制对称破缺插值的各种自旋玻璃模型，如感知器、Hopfield、稀释p-自旋和p-sat模型。在所有这些模型中，这种插值形式地再现了理论物理学家预测的解，但由于谢林顿-柯克帕特里克模型中Parisi公式的证明方法并不直接适用于这些模型，因此需要找到新的方法来控制这些插值中的误差项。此外，该提案还包括关于球面Sherrington-Kirkpatrick模型中重叠的联合分布、Parisi泛函的性质以及通过Almeida-Thouless线表征Sherrington-Kirkpatrick模型中的复制对称区域的其他几个问题。统计力学中的几个模型，称为平均场自旋玻璃模型，最初是由理论物理学家介绍和研究的，他们发展了一个令人印象深刻的启发式理论，对这些模型的行为给出了详细的预测，并影响了许多其他领域的研究，远远超出了最初问题的范围。对这位物理学家的一些预测进行严格的数学证明，需要一些新的想法和方法，这些想法和方法可能在概率、统计物理、计算机科学和统计学等其他领域有用。目前的建议将继续研究几个有前途的方向。

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.