Random Matrix Theory and Its Applications

RANDOM MATRIX THEORY AND ITS APPLICATIONS

Spring 2009
Dr. Eitan Sayag

Meetings: TBA (4 hours: most likely on Monday and Wednesday).
Office hours: TBA or by appointment

Instructor: Eitan Sayag
Office: 107, Math Building N58.
Phone: 7-2713
Email:
Course description:
The course is an introduction to the mathematical theory of random matrices and its applications to various scientific disciplines. We will cover results on the asymptotic properties of various random matrix models (Wigner matrices, Gaussian ensembles, Dyson's beta-ensemble).. We will learn about asymptotic phenomenon (eg concentration of measures) and about mathematical tools such as Stieltjes transform and Free Probability used to characterize infinite random matrices.

We will provide many real life applications of the theory especially to wireless communications, computer science and stability questions of linear systems.

Tentative Plan.

·  Overview of the course and some History of RMT

·  Review: Probability and convergence of measures.

·  Review: Central limit theorem and concentration of measure.

·  Review: Eigenvalues, Singular values and their applications.

·  Classic random matrix ensembles (GOE, GUE, GSE)

·  Formulation of the classical results of RMT.

·  Stieltjes Transform Method.

·  Wigner's semicircle law

·  Marchenko-Pastur law

·  Orthogonal polynomials

·  What is Free probability and its relation to RMT.

·  Stability: When numerical Analysis meets random matrix theory

·  MIMO channels: asymptotic analysis of capacity, design of receivers using Random matrix theory

·  Applications of Random matrices to Data Analysis.

·  Relation to Random graphs and Sensors Networks.

Prerequisites: Proficiency in linear algebra and basic probability will be a plus although I intend to review quickly whatever notions and properties I will need. A familiarity with MATLAB will also be useful.

For students of the mathematics department the courses: Probability (201.1.8001), Calculus 3 (201.1.0031), Algebra 2 (201.1.702.1).

For EE students: Probabilty Theory for EE (201.1.9831), Calculus 2 for EE (201.1.9821), Linear algebra for EE (201.1.9851).

Evaluation: There will be biweekly exercise sheet (30%) a final project – presenting a paper on the subject matter of the course (40%) and an independent work in class based on the exercises and the material taught in class. This work will be in the final meeting (30%).

Recommended Books

A couple of useful references:

·  M. Mehta: Random Matrices.

·  G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to Random Matrices (to appear)
available from O. Zeitouni's webpage

·  A. Guionnet: Large random matrices: lectures on macroscopic asymptotics. Lectures from the 36th Probability Summer School held in Saint-Flour, 2006.
available from the author's webpage