CAREER: Random Matrices, Random Schroedinger and Communication

这个项目在随机矩阵理论(RMT)中承担了三个不同的研究领域。第一个起源于研究人员和合作者最近发现了RMT的特雷西-维多姆定律和一族随机薛定谔算符之间的联系，这些定律是对所谓的“贝塔系综”的推广。这种联系为更深入地理解特雷西-维多姆定律的分析结构打开了大门。给出了一种新的方法来研究随机矩阵和随机薛定谔在谱边缘的普适性问题。第二组问题涉及不具有假设对称性的矩阵系综的整体涨落。由于这些涨落与高斯自由场之间的联系，这些问题在理论物理学中越来越引起人们的兴趣。最后，研究人员展望了RMT在无线技术中的几个应用，其中RMT的概率和分析技术被用于估计具有反馈的多天线通信系统的容量和性能。矩阵及其谱(特征值)是贯穿整个数学及其应用的基本对象。对随机矩阵(RMT)的研究解决了一个自然的问题：“典型矩阵的特征值是什么样子的？”或者，另一种说法是：“光谱的哪些性质是通用的，因为它们只取决于矩阵系综的粗略特征(例如对称性)？”因此，RMT在从统计学到数论再到算子代数的不同数学领域以及在物理学和工程学中产生严重影响也就不足为奇了。在理论方面，本课题为研究对称矩阵的最大特征值和非对称矩阵的整体(或通用)特征值的普适性质提供了新的思路。在更直接的应用方面，这个项目着眼于无线技术理论中的问题，有趣的是，许多高效通信系统的设计和性能分析导致了RMT语言的数学问题。这里的工程问题具有极其重要的实际意义--虽然对无线通信的需求持续增长，但频谱仍然是一种有限的资源。这个项目为数学家和电气工程师之间的合作提供了一个自然的环境，该项目将与共同指导的研究生和博士后研究人员一起主办联合会议。

This project takes on three distinct areas of research in Random Matrix Theory (RMT). The first stems from the recent discovery by the investigator and collaborators of a connection between the Tracy-Widom laws of RMT, their generalization to the so called "beta-ensembles", and a family of random Schroedinger operators. This connection opens the door to a deeper understanding of the analytic structure of Tracy-Widom laws. It also presents a new method for investigating the universality question at the spectral edge for both random matrices and random Schroedinger.The second set of problems concerns bulk fluctuations for ensembles of matrices with no assumed symmetry. Such questions are of growing interest in theoretical physics on account of the ties between these fluctuations and the Gaussian Free Field. Finally, the investigator looks at several applications of RMT to wireless technology where probabilistic and analytic techniques from RMT are employed to estimate the capacity and performance of multiple-antenna communication systems with feedback.Matrices and their spectra (eigenvalues) are fundamental objects throughout mathematics and its applications. The study of random matrices (again, RMT) addresses the natural question: "What do the eigenvalues of the typical matrix look like?" Or, said another way: "What properties of the spectra are universal in that they depend only on the rough characteristics (e.g. symmetries) of the matrix ensemble?" It is therefore not surprising that RMT has serious impact in disparate areas of mathematics ranging from Statistics to Number Theory to Operator Algebras, as well as in Physics and Engineering. On the theoretical front, this project brings new ideas to the study of the universal properties of the largest eigenvalues in symmetric matrices and also the bulk (or generic) eigenvalues in matrices without symmetry. On the more directly applied side, this project looks at problems in the theory of wireless technology where, interestingly, the design and performance analysis of many efficient communication systems leads to mathematical problems cast in the language of RMT. The engineering issues here are of the utmost practical importance - while the demand for wireless communications continues to grow, the spectrum remains a limited resource. Presenting a natural setting for collaborations between mathematicians and electrical engineers, this project will sponsor joint meetings along with jointly mentored graduate students and postdoctoral researchers.