Mathematical Sciences: Integrable Models in Mathematics and Physics

数学科学：数学和物理中的可积模型

• 批准号: 9303413
• 负责人: Craig Tracy
• 金额: $ 22.5万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303413&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Integrable; Models; Mathematics

This research project develops new methods involving completely integrable differential equations (of the Painleve type), operator theory and asymptotic analysis to study the level spacing distributions for various random matrix ensembles. The level spacing distribution for various orthogonal polynomial ensembles can be expressed in terms of a Fredholm determinant. The methods of the PI and his collaborator Harold Widom show that this determinant is also a tau-function. In the general problem that will be analyzed the tau function has two distinct types of variables. There are the ``KP type variables'' which describe a KP flow on the Sato Grassmannian and there are the ``deformation type variables.'' It is the combination and interplay between these two types of variables that give the general tau-function. Particular examples of this general tau function are important in disordered conductors, numerical analysis associated with the condition number of a matrix, in addition to the standard applications of random matrix ensembles to the analysis of eigenvalue statistics. The methods developed in this program will give detailed and explicit formulas for these important cases. %%% Complex systems that arise in nuclear physics, atomic and molecular physics, condensed matter physics dealing with media with impurities require a type of mathematics that gives results for averaged quantities, since on the macroscopic everyday world the variables that are controlled in various experiments (like concentration of an impurity, the energy of the system) do not completely specify the microscopic system. Therefore one develops statistical methods that predict average behavior or give the probabilities of deviation from this average behavior. The subject of random matrices is one such statistical theory that has been successfully applied to the above problems in physics along with a host of more theoretical problems in mathematics and practical problems in numerical analysis. This research project develops new methods using the tools of differential equations to study these statistical theories.

该研究项目开发了涉及完全可以整合的微分方程（Painleve类型），操作员理论和渐近分析的新方法，以研究各种随机矩阵集合的水平间距分布。 各种正交多项式集合的水平间距分布可以用弗雷德姆的决定因素表示。 PI和他的合作者Harold Widom的方法表明，这种决定因素也是Tau功能。 在将要分析的一般问题中，tau函数具有两种不同类型的变量。有``KP类型变量''描述了Sato Grassmannian上的KP流动，并且有``变形类型变量''。这是这两种类型的变量之间的组合和相互作用，从而使一般的Tau功能函数提供了。 此通用TAU功能的特定示例在无序导体中很重要，除了随机矩阵集合在特征值统计分析中的标准应用外，与矩阵的状况数量相关的数值分析。 该程序中开发的方法将为这些重要情况提供详细而明确的公式。 在核物理学，原子和分子物理学，与具有杂质的媒体打交道的物理学中出现的复杂系统需要一种数学，从而为平均数量提供结果，因为在宏观的日常世界中，在各种实验中控制的变量在各种实验中受控（例如，构成杂质的浓度，也不是完全属于系统的杂货）。 因此，人们开发了预测平均行为或给出偏离这种平均行为的概率的统计方法。 随机矩阵的主题是一种这样的统计理论，该理论已成功地应用于物理学的上述问题，以及数值分析中数学和实际问题的许多理论问题。 该研究项目使用微分方程工具开发了新方法来研究这些统计理论。