Research in Random Matrices and Integrable Systems

随机矩阵和可积系统研究

• 批准号: 0243982
• 负责人: Harold Widom
• 金额: $ 12.6万
• 依托单位: University of California-Santa Cruz
• 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=0243982&HistoricalAwards=false
• 关键词: Research; Random; Matrices; Integrable; Systems

PI: Harold Widom, UC Santa CruzDMS-0243982*********************************************************************There are four projects, the first three joint with Craig A. Tracy and the fourth joint with Estelle L. Basor. The first is to search for a system of partial differential equations for the finite-dimensional distribution functions for the Airy process, which is expected to describe all growth processes in the KPZ universality class. (This has already been accomplished.) The second project is to find limit theorems for the Hall-Littlewood polynomials, interpolating known results for the Schur functions (Johansson) and the shifted Schur functions (Tracy and the PI). New methods are needed here because of a lack of determinantal representation. The third project is to complete earlier work on the asymptotics of solutions to the periodic Toda equations by determining the asymptotics on the critical curves, where the asymptotics will take a very different form. The fourth project is to prove correct the conjectured constant in an asymptotic expansion of Dyson related to the spacings between eigenvalues of random matrices in the GUE universality class. To this end it is hoped to refine the method recently used by Basor and the PI to obtain the asymptotics of Wiener-Hopf determinants with Fisher-Hartwig symbols. Many physical systems possess such complicated behavior that exact predictions become impossible. Random matrix theory provides mathematical models that allow a simulation of such behavior and predictions that allow comparison with experiment. This was its original motivation, but it has since had significant applications in other areas of mathematics, science and technology. Work of the PI (largely collaborative) in this subject has had application in statistics where it has been applied to the analysis of large data sets, in communications where it has been used to study the start-up behavior of a long production line or the transient flow of messages over a long path in a communications network, and in physics and technology where it has had an impact on the theory of gap fluctuations in a metal grain or quantum dot. This project is concerned with different aspects of the theory including the search for the solution to unsolved problems. The project is also concerned with the connection of random matrix theory withcombinatorics and growth processes. This is a more recent development that could shed light on the question of universality of probability distributions related to the Airy process mentioned above.

PI：Harold Widom，加州大学圣克鲁斯分校DMS-0243982******************************************** ******************************共有四个项目，前三个与 Craig A. Tracy 合作，第四个与 Estelle L 合作巴索尔。第一个是寻找艾里过程的有限维分布函数的偏微分方程组，该系统有望描述 KPZ 普适类中的所有生长过程。 （这已经完成。）第二个项目是找到 Hall-Littlewood 多项式的极限定理，对 Schur 函数（Johansson）和移位 Schur 函数（Tracy 和 PI）的已知结果进行插值。由于缺乏决定性的表示，这里需要新的方法。第三个项目是通过确定临界曲线上的渐近性来完成周期 Toda 方程解的渐近性的早期工作，其中渐近性将采取非常不同的形式。第四个项目是证明戴森渐近展开式中与 GUE 普适类中随机矩阵特征值之间的间距相关的猜想常数的正确性。为此，希望改进 Basor 和 PI 最近使用的方法，以获得具有 Fisher-Hartwig 符号的 Wiener-Hopf 行列式的渐近性。许多物理系统具有如此复杂的行为，以至于精确的预测变得不可能。随机矩阵理论提供了数学模型，可以模拟此类行为和预测，从而可以与实验进行比较。这是其最初的动机，但此后它在数学、科学和技术的其他领域有了重要的应用。 PI（主要是协作）在该主题中的工作已在统计学中得到应用，其中已应用于大数据集的分析，在通信中已被用于研究长生产线的启动行为或通信网络中长路径上的瞬态消息流，以及物理和技术中的瞬态流，它对金属颗粒或量子点中的间隙波动理论产生了影响。该项目涉及理论的不同方面，包括寻找未解决问题的解决方案。该项目还涉及随机矩阵理论与组合学和增长过程的联系。这是最近的进展，可以阐明与上述艾里过程相关的概率分布的普遍性问题。