Probability Theory and Spin Glasses.

概率论和自旋玻璃。

• 批准号: 0243813
• 负责人: Michel Talagrand
• 金额: $ 19.14万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0243813&HistoricalAwards=false
• 关键词: Probability; Theory; Spin; Glasses.

0243813Talagrand Mean field models for spin glasses (such as the famous Sherrington Kirkpatrick model) exhibit some completely new phenomenon of probability theory, that at a high level can be described as the existence of intricate structures among the near extreme values of large families of correlated random variables. The rigorous study of these models is still in its infancy, and the main goal of this proposal is to pursue it. Two of the specific goals are to obtain a better understanding of the low-temperature phase of the p-spin interaction model, and to start the study of the new Hamiltonians introduced by F. Guerra in his broken replica-symmetry bound for the Sherrington Kirkpatrick model. One of the central themes of probability theory is the emergence of a kind of order out of large number of random events. This is expressed by classical theorems, such as the law of large numbers. These classical results relate to situations where the random events are independent, that is, the outcome of one does not influence the outcome of the others. Such an assumption is unrealistic in practice. We investigate the emergence of new types of collective behavior in large collections of random events, where each event can, and will, influence the outcome of the others. The specific mathematical models have been motivated by the behavior of magnetized particles, and are known as "spin glasses".

0243813自旋玻璃的Talagrand平均场模型(如著名的Sherrington Kirkpatrick模型)展示了一些全新的概率论现象，在更高的水平上，可以描述为在大族相关随机变量的近极值中存在复杂的结构。对这些模型的严格研究还处于初级阶段，这项提案的主要目标是追求它。其中两个具体目标是更好地理解p自旋相互作用模型的低温相，并开始研究F.Guera在他对Sherrington Kirkpatrick模型的破碎复制对称界中引入的新哈密顿量。概率论的中心主题之一是在大量随机事件中出现一种秩序。这是用经典的定理来表达的，比如大数定律。这些经典结果涉及随机事件相互独立的情况，即一个事件的结果不影响其他事件的结果。这样的假设在实践中是不现实的。我们调查了在随机事件的大型集合中出现的新类型的集体行为，其中每个事件都可以并将影响其他事件的结果。具体的数学模型是由磁化粒子的行为驱动的，被称为“自旋玻璃”。