CAREER: Random Matrices, Random Schroedinger and Communication

该项目采用随机矩阵理论（RMT）中三个不同的研究领域。第一个源于研究者和合作者最近发现的RMT的tracey - widom定律之间的联系，他们将其推广到所谓的“beta-ensembles”，以及随机薛定谔算子家族。这种联系为更深入地理解特蕾西-威姆定律的分析结构打开了大门。给出了一种研究随机矩阵和随机薛定谔谱边普适性问题的新方法。第二组问题涉及没有假设对称性的矩阵系综的体波动。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引起了越来越大的兴趣。最后，研究者研究了RMT在无线技术中的几种应用，其中RMT的概率和分析技术被用来估计带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）是贯穿数学及其应用的基本对象。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，用另一种方式说：“光谱的什么性质是普遍的，因为它们只依赖于矩阵系综的粗糙特征（例如对称性）？”因此，RMT在数学的不同领域（从统计学到数论到算子代数）以及物理和工程中产生严重影响并不奇怪。在理论方面，本项目为对称矩阵中最大特征值的普适性质的研究以及非对称矩阵中本体（或一般）特征值的研究带来了新的思路。在更直接的应用方面，该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用RMT语言表达的数学问题。这里的工程问题具有极大的实际重要性——尽管对无线通信的需求持续增长，但频谱仍然是有限的资源。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，该项目将赞助联合会议以及共同指导的研究生和博士后研究人员。

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.