Random Matrices (0366-4646)

Fall 2017, Tel Aviv University

Location: Dan David 204, Tuesdays 10-13

Instructor: Ron Peled

The course is an introduction to random matrix theory.

Syllabus

The main topic of the course is the distribution of eigenvalues and eigenvectors of random matrices. This field emerged from applications in statistics (data analysis) and physics (energy levels of heavy nuclei) and has developed into a modern, and vast, mathematical field. We will discuss the basic results pertaining to Wigner random matrices (symmetric matrices with independent entries) and focus on the Gaussian ensembles GOE, GUE which possess special symmetries. Among the topics to be discussed, time permitting, are Wigner's semicircle law, the distribution of eigenvalues on the microscopic scale and the Tracy-Widom distribution for the maximal eigenvalue. Potential additional topics include random operators (such as the random Schrödinger operator) and their localization phenomena, and applications of random matrix theory to the study of the longest increasing subsequence of a uniformly random permutation.
Prerequisites: Probability for Mathematicians or Probability for Sciences, Complex Function Theory and Calculus 3.

Books

We will mostly follow the book "An Introduction to Random Matrices" by Anderson, Guionnet, Zeitouni.
Other relevant books include:
Terence Tau, Topics in random matrix theory,
Pastur, Shcherbina, Eigenvalue Distribution of Large Random Matrices,
Forrester, Log-Gases and Random Matrices,
Mehta, Random matrices,
Aizenman, Warzel, Random Operators: Disorder Effects on Quantum Spectra and Dynamics,
Baik, Deift, Suidan, Combinatorics and Random Matrix Theory.

Exercises

Homework exercise 1. To be handed in by December 19 in class.