506

MTH 506

Overview

E. Wigner used a random matrix to model the statistical behavior of the energy resonances of neutron scattering experiments in the 1950sk The model was remarkably successful. Since then, connections to diverse fields such as the zeros and moments of the Riemann zeta function (H. Montgomery, F. Dyson, and others), longest increasing sequences of permutations (J. Baik, P. Deift, J. Johansson), and bus schedules in Mexico have been found. Most if not all of these connections have defied both heuristic and rigorous explanation. The wikipedia page on random matrices lists a few more of these conections.

We will focus first on the limiting spectral distributions that appear in some basic families of random matrices: Wigner’s semicircular law, the Marcenko-Pastur distribution and the “disc” law. These distributions are universal under some very mild assumptions; i.e., they don’t care about the particulars distribution of the random variables that you put into the random matrix. Then, we will cover topics from free probability, the Gaussian ensembles, and briefly discuss Dyson Brownian motion.

Prerequisites: MTH 403 Theory of Probability or equivalent. MTH 504 Stochastic Processes is not required, but it will help. I will spend the first few lectures covering some of the tools we need.

Grading

3 grades, A, A-, B. Based on in-class presentation. Need 2/3rds of presentations to be satisfactory for A. 1/3rd to be satisfactory for A-. B is almost never awarded unless you don’t participate or stop showing up.

Textbook

  1. Topics in Random Matrix Theory, by Terence Tao. Available online on Terry’s website. We will start from chapter 2 and more or less follow this
  2. Around the circular law, by Bordenave and Chafai. Downloadable from the web.
  3. Random Matrices, Madan Lal Mehta.

Notes

They are in svg or svgz format. Can be read easily in the free cross-platform program Write. Sometimes, I might use the occasional xopp file, which can be read using xournal++.

  1. introduction
  2. Tao 1.3 spectral theorem, and eigenvalue inequalities