Courses:

This is a course on the mathematics and applications of infinite random matrices. We will learn about the tools such as the Stieltjes transform, and Free Probability used to characterize infinite random matrices. Our emphasis will be on exploring known connections between these tools (such as the combinatorial aspects of free probability) and discovering new connections (such as between multivariate orthogonal polynomials and free cumulants of free probability).

Our aim is to touch upon various branches of the study of infinite random matrices-a consequence is that we will end up lingering on some areas longer than others. Our hope is that this course will confer:

• Some familiarity with several of the main thrusts of work in infinite random matrices-sufficient to give you some context for formulating and seeking known solutions to applications in engineering and physics;
  
  
• Sufficient background and facility to let you read current research publications in the area of infinite random matrix theory;
  
  
• A set of tools, both analytical and computational, for the analysis of new random matrices that arise in new problems you may encounter.

Prerequisites

No particular prerequisites are needed. We assume that students have had an undergraduate course in Linear Algebra (18.06) or its equivalent and some exposure to probability (6.041 or 6.042J are more than sufficient). Knowledge of combinatorial theory is a bonus. A familiarity with MATLAB® will also be useful.

Content

The goal for the course is, paradoxically, to be broad as well as deep. Our plan is to touch upon the following broad areas while attempting to uncover deep insights into the underlying mechanisms that unify these areas.

Below is a tentative list of topics that might be covered in the course; We will select material adaptively based on student background, interests, and rate of progress. If you are interested in some other topics, please let us know and we'd be happy to accommodate your interests.

• Combinatorial Aspects: Using combinatorial techniques to derive the limiting distribution of the three classical random matrix ensembles. Path counting and random matrix theory. Generalizations to counting paths on torii.
  
  
• Stieljtes Transform Based Methods: The Mar¡cenko-Pastur theorem. Other generalizations. Silverstein's sample covariance matrix. Convergence issues.
  
  
• Free Probability: The concept of freeness. Free cumulants and non-crossing partitions. The R and S transform. Fluctuations and Second Order Freeness. Combinatorial interpretations.
  
  
• Equilibrium Measure: The Hermite, Laguerre and Jacobi orthogonal polynomials. Interpretation of the limiting distribution as the equilibrium measure of (univariate) orthogonal polynomials. Applications to physics.
  
  
• Fredholm Determinants: Tracy-Widom Distribution. Eigenvalue spacings and the Riemann Hypothesis.
  
  
• Jack Polynomials: Multivariate orthogonal polynomials. Combinatorial aspects. Connections to random matrices.
  
  
• Applications: Wireless Communications, Statistical Physics.

Homework

Homework assignments will be handed out bi-weekly. They will mainly consist of MATLAB® based explorations of the material covered in class. You will not need to turn them in, although being able to do them will greatly help your understanding of the material.

Mid-Term Project

You will be asked to read a paper on a topic of interest to you that involves random matrix theory. See the project section for more information.

Semester Project

The semester project can be an extension of the mid-term project if it sustains your interest. Otherwise, you will be asked to come up with some insights into a random matrix problem that is of interest to you.

See the project section for more information.

Grading

Since this is an advanced graduate class on a very active research area, the grading will be based on your participation in the class.

Textbooks

There are no textbooks covering a majority portion of the material we will be studying in this course. Please see the readings section for more information.