Random matrix theory is extremely rich and diverse, bringing together topics and tools from physics, probability, linear algebra, analysis, statistics, etc., and providing links between fields as far apart as number theory and computational biology, or statistical mechanics and wireless communications.

Random matrices are characterized by the fact that their entries are variables with prescribed distributions (often independent and identically distributed; e.g. standard normal N(0,1)). The objects of study in random matrix theory are statistics of eigenvalues, otherwise known as eigenstatistics. Such objects include moments of the empirical distribution (i.e. E[tr(A^k)], average traces of powers of the random matrix A), moments of the determinant, asymptotics of the empirical distribution, global and local fluctuations, etc.

This course aims to take a look at the interplay between combinatorics and random matrix theory--more specifically, at combinatorial structures that appear naturally in (asymptotics of) eigenstatistics of random matrices. Such objects include Dyck and Motzkin paths, maps over surfaces of genus 2, Catalan numbers, Jack polynomials, etc. Other interesting structures that we will encounter are tridiagonal matrices, uni- and multivariate orthogonal polynomials, random recurrences, and more.

The course is intended primarily, but not exclusively, for first and second year graduate students, and will consist of a combination of lectures (in the beginning) and student presentations (toward the end of the quarter). Aside from a presentation based on reading material that the instructor will provide, students will be expected (and encouraged) to contribute to class discussions.

The following two figures constitute illustrations of topics that will be approached and explored in this course: the semicircle law and a duality principle for averages of Jack polynomials.

Texts: A collection of papers and book extracts (will be provided by the instructor).

Prerequisites: No per se prerequisites, but basic familiarity with Linear Algebra, Combinatorics, and Probability (as provided, for example, by an upper level undergraduate course in each of the topics) would help. Some experience with MATLAB or Maple would be useful, but by no means necessary.