Mean-Field Spin Glass Models

平均场自旋玻璃模型

• 批准号: 0904565
• 负责人: Dmitriy Panchenko
• 金额: $ 14.68万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0904565&HistoricalAwards=false
• 关键词: Mean; Field; Spin; Glass; Models

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.

该奖项是根据2009年《美国复苏和再投资法案》(Public Law 111-5)资助的。在Francesco Guera发现复制对称破缺插值法和Michel TALAGRAND证明著名的Parisi自由能公式之后，在过去几年中，人们对场自旋玻璃模型，特别是Sherrington-Kirkpatrick模型有了更好的理解。目前的提案包括几个研究方向，这些方向将试图在最近取得的进展的基础上再接再厉。一个项目建议研究由Gibbs测度的某种随机稳定性产生的关于重叠分布的Ghirlanda-Guera恒等式是否暗示Parisi超度量猜想。另一个项目涉及用于各种自旋玻璃模型的Guera副本对称破缺内插的一些自然类似物，例如感知器、Hopfield、稀释p-自旋和p-sat模型。在所有这些模型中，这样的内插形式上复制了理论物理学家预测的解，但由于Sherrington-Kirkpatrick模型中Parisi公式的证明方法不直接适用于这些模型，所以需要寻找新的方法来控制这些内插中的误差项。此外，该提案还包括关于球形Sherrington-Kirkpatrick模型中重叠的联合分布、Parisi泛函的性质以及通过Almeida-Thouless线描述Sherrington-Kirkpatrick模型中副本对称区的其他几个问题。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引入和研究的，他们发展了一个令人印象深刻的启发式理论，给出了关于这些模型的行为的详细预测，并影响了许多其他领域的研究，远远超出了原始问题的范围。要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，这些想法和方法很可能在概率论、统计物理、计算机科学和统计学的其他领域有用。目前的提议将继续在几个有希望的方向上进行研究。