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年的《美国回收与再投资法》（公法111-5）资助的。在过去的几年中，在过去的几年中，在过去几年中，更好地理解了Sherrington-kirkpatrick模型在Francesco guerra和庆祝Parister prifice prification free parter the free parter the prificate interage interpolation之后，更好地理解了Sherrington-Kirkpatrick模型。当前的提案包括一些研究方向，这些方向将试图基于最近的进步。一个项目提议研究ghirlanda-guerra对重叠分布的身份是否是由吉布斯（Gibbs）衡量的某些随机稳定性产生的，这意味着巴黎超级估计。另一个项目涉及各种自旋玻璃模型（例如Perceptron，Hopfield，稀释的P-Spin和P-SAT模型）的Guerra复制对称性破坏插值的许多自然类似物。在所有这些模型中，这种插值正式重现了理论物理学家预测的解决方案，但是由于Sherrington-Kirkpatrick模型中巴黎公式的证据的方法论并不直接适用于这些模型，因此需要查找这些插入中误差项的新方法。此外，该提案还包括有关在球形Sherrington-Kirkpatrick模型中重叠的联合分布的其他几个问题，Parisi功能的属性以及复制品对称区域的表征在Sherrington-Kirkpatrick中通过Almeida-thouth Thouthoune-thouth-thoutield Modelity在Sherrington-kirkpatrick中的统一模型。建立了令人印象深刻的启发式理论的物理学家，对这些模型的行为做出了详细的预测，并影响了许多其他研究领域，这远远超出了原始问题的范围。某些物理学家预测的严格数学证明需要许多新的想法和方法，这些想法和方法可能在其他概率，统计物理，计算机科学和统计的领域中有用。当前的建议将继续以几个有希望的方向进行研究。