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积微分方程（Painleve 型）、算子理论和渐近分析的新方法，以研究各种随机矩阵系综的能级间距分布。 各种正交多项式系综的电平间距分布可以用 Fredholm 行列式来表示。 PI 和他的合作者 Harold Widom 的方法表明，这个行列式也是一个 tau 函数。 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。有描述 Sato Grassmannian 上的 KP 流的“KP 型变量”和“变形型变量”。正是这两种类型变量之间的组合和相互作用给出了一般 tau 函数。 除了随机矩阵系综在特征值统计分析中的标准应用之外，这种通用 tau 函数的特定示例在无序导体、与矩阵条件数相关的数值分析中也很重要。 该计划中开发的方法将为这些重要情况提供详细而明确的公式。 %%% 核物理、原子和分子物理、凝聚态物理中出现的涉及杂质介质的复杂系统需要一种能够给出平均量结果的数学，因为在宏观的日常世界中，各种实验中控制的变量（如杂质的浓度、系统的能量）并不能完全指定微观系统。 因此，人们开发了预测平均行为或给出偏离该平均行为的概率的统计方法。 随机矩阵主题就是这样一种统计理论，它已成功应用于解决上述物理问题以及许多数学中的理论问题和数值分析中的实际问题。 该研究项目开发了使用微分方程工具来研究这些统计理论的新方法。