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.
This project focussed on the study of random matrices, in particular those quantities (typically the spectrum, or eigenvalues) connected to a broad class of matrices whose behavior is universal in the limit of large dimension. This universality phenomena can be compared with the classical central limit theorem in which the normalized sum of a large number of independent observations (random variables) takes a Gaussian (bell-curve) distribution in the limit. As the Gaussian is ubiquitous throughout mathematics and science, increasingly so too are the distributions arising in random matrix theory. For instance, the Tracy-Widom distribution, first discovered in the description of the largest eigenvalue for certain symmetric matrices, is now understood to capture the fluctuations in basic models in combinatorics, statistical mechanics, and mathematical statistics. A chief part of the PI's research funded under this project has resulted in new formulations of the Tracy-Widom (and related) laws connected with random partial differential equations. These new descriptions provide an additional set of tools with which to study the basic properties of these fundamental distributions. In a separate direction, this project explored direct applications of random matrix theory to wireless communications. The connection here is as follows. The capacity for various models of communications are given in terms of (increasingly complicated) functions of a "channel matrix" which captures the qualitative properties of the path between transmitters and receivers of the underlying signal. As can be imagined, these path properties are quite challenging to model effectively (especially when dealing with communictaions among mobile devices), and one is left to try to capture the "typical", or random, channel. In this way, important benchmarks of state of the art communication systems are described by functionals of certain random matrices. This project funded in part related research of one joint math/electrical engineering Ph.D. student, as well as a 2008 conference which brought together mathematicians, physicists, and engineers working in these important applications of random matrix theory.
