CAREER: Random Matrices, Random Schroedinger and Communication

职业：随机矩阵、随机薛定谔和通信

• 批准号: 0645756
• 负责人: Brian Rider
• 金额: $ 40万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0645756&HistoricalAwards=false
• 关键词: CAREER; Random; Matrices; Schroedinger; Communication

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.

该项目涉及随机矩阵理论（RMT）的三个不同研究领域。 第一个源于最近发现的调查员和合作者的Tracy-Widom法律之间的联系RMT，他们的推广到所谓的“β-系综”，和一个家庭的随机薛定谔算子。 这种联系打开了大门，以更深入地了解特雷西-Widom法律的分析结构。它还提出了一种新的方法来调查的普遍性问题在频谱边缘的随机矩阵和随机Schroedinger。第二组问题涉及散装波动的矩阵合奏没有假设的对称性。 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，研究人员着眼于几个应用RMT无线技术的概率和分析技术，从RMT被用来估计的容量和性能的多天线通信系统的feedback. Matrix和他们的频谱（特征值）是整个数学和它的应用的基本对象。 对随机矩阵的研究（同样是RMT）解决了一个自然的问题：“典型矩阵的特征值是什么样的？”或者，换句话说：“光谱的什么性质是普适的，因为它们只依赖于矩阵系综的粗略特征（例如对称性）？”“因此，RMT在从统计到数论到算子代数以及物理和工程等不同的数学领域产生严重影响也就不足为奇了。 在理论方面，该项目为研究对称矩阵中最大特征值的普适性质以及不对称矩阵中的批量（或通用）特征值带来了新的思路。 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RMT语言中的数学问题。 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，该项目将赞助沿着联合指导的研究生和博士后研究人员的联合会议。