Some problems on mean-field spin glasses

平均场自旋玻璃的若干问题

• 批准号: 1513605
• 负责人: Wei-Kuo Chen
• 金额: $ 13.5万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1513605&HistoricalAwards=false
• 关键词: Some; problems; mean; field; spin

Spin glasses are disordered spin systems invented by theoretical physicists in the early 70's. The aim is to describe the strange magnetic behaviors of certain alloys. Over the past decades, physicists' investigation has generated a series of methodologies and predictions. In the community of mathematics, spin glass models are purely probabilistic objects that present highly complex random structures. The general aim of this research project is to pursue mathematical investigation of physical phenomena including the complexity of the random Hamiltonian, the structure of the Gibbs measure and the thermodynamic limit of the free energy. As well as being important in the field of probability, the tools and results obtained in this research plan are relevant to a variety of scientific branches including computer science, theoretical biology and social networks. More precisely, this proposal is concerned about the Sherrington-Kirkpatrick (SK) model. The PI will investigate the structure and the role of the functional order parameter in the low temperature regime. In addition, the conjectured ultrametricity of the Gibbs measure and the problem of chaos in temperature will be analyzed following the Guerra-Talagrand replica symmetry breaking bound. The PI will as well study the central limit theorem for some physical quantities such as the overlap and magnetization especially at the critical temperature using Stein's method. Another direction is the SK model with heavy tail disorder. As now higher order interactions are allowed, the system has been predicted to exhibit unusual spin glass phenomena. The last part of the project involves the bipartite version of the SK model, which was used to model collective properties of two interactive large groups of individuals motivated by the studies of the social and neural networks, biology and economics. The PI will be concentrated on the computation of the limiting free energy.

旋转眼镜是由理论物理学家在70年代初发明的无序旋转系统。目的是描述某些合金的奇怪磁性行为。在过去的几十年中，物理学家的研究产生了一系列方法和预测。在数学社区中，自旋玻璃模型是纯粹的概率对象，它们具有高度复杂的随机结构。该研究项目的总体目的是对物理现象进行数学研究，包括随机哈密顿量的复杂性，吉布斯的结构测量和自由能的热力学极限。除了在概率领域很重要外，本研究计划中获得的工具和结果与包括计算机科学，理论生物学和社交网络在内的各种科学分支有关。更确切地说，该提议关注Sherrington-Kirkpatrick（SK）模型。 PI将研究功能顺序参数在低温方向上的结构和作用。此外，将在Guerra-Talagrand副本对称性断裂结合后分析Gibbs度量的猜想的超级衡量和混乱问题。 PI还将研究一些物理量的中心极限定理，例如重叠和磁化，尤其是在使用Stein方法的临界温度下。另一个方向是具有重型尾部障碍的SK模型。由于现在允许高阶相互作用，该系统被预测表现出异常的自旋玻璃现象。该项目的最后一部分涉及SK模型的两部分版本，该版本用于模拟由社会和神经网络，生物学和经济学的研究激励的两个交互式大型个人的集体特性。 PI将集中于限制自由能的计算。