Research in Random Matrices and Integrable Systems

主要研究者：哈罗德·维多姆，加州大学圣克鲁斯分校DMS-0243982**特蕾西和第四个关节与埃斯特尔L。巴索首先是寻找一个系统的偏微分方程的有限维分布函数的艾里过程，这是预期描述所有的增长过程中的KPZ普适类。(This已经完成）。第二个项目是找到极限定理的霍尔-Littlewood多项式，插值已知的结果舒尔函数（约翰松）和转移舒尔函数（特雷西和PI）。由于缺乏行列式表示，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早期关于周期户田方程解的渐近性的工作，其中渐近性将采取非常不同的形式。第四个项目是证明GUE普适类中与随机矩阵特征值间距有关的Dyson渐近展开式中的固定常数是正确的。为此，希望改进最近使用的方法Basor和PI获得渐近的维纳-霍普夫行列式与费舍尔-哈特维希符号。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允许模拟这种行为的数学模型和允许与实验进行比较的预测。这是它最初的动机，但它在数学，科学和技术的其他领域有重要的应用。PI的工作（主要是协作的）在这个主题中已经在统计学中应用，其中它已经被应用于大型数据集的分析，在通信中，它已经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消息的瞬时流，在物理学和技术领域，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，包括寻求解决未解决问题的方法。该项目还涉及随机矩阵理论与组合学和增长过程的联系。这是一个最近的发展，可以阐明的问题的普遍性概率分布有关的艾里过程上述。

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.