Free Probability and Ramanujan Graphs

Spring 2024

About the course

Free probability is a mathematical theory that generalizes classical probability theory to the non-commutative setting. In the classical setting, random variables are often combined and studied using their joint distributions, which take into account the dependencies and correlations between them. In free probability, "freeness" is an abstract concept that describes a lack of correlations between non-commutative random variables.

Free probability theory is an exciting and growing field with deep connections to random matrix theory, quantum mechanics, and other areas. Closer to home, free probability, or more precisely a finite analog of it, was employed in a seminal sequence of works for the construction of bipartite Ramanujan graphs.

The mathematics of free probability is quite sophisticated, involving advanced concepts from functional analysis, operator algebras, and combinatorics. In this course, we will embark on an exploration of free probability theory, including both its infinite and finite manifestations, with a particular emphasis on applications in graph theory.

Technicalities

When and where

Lectures are on Thursdays 10:10-13:00 and recitations by Gal Maor are right after, namely, Thursdays 13:10-14:00. Location TBD.

Grade

Half of the grade will be based on about 6 homework assignments. Homework submission will be in pairs if the the number of participants will be high enough. The other half of the grade will be derived from a take home exam.

Literature

We will mostly follow the great book Lectures on the Combinatorics of Free Probability Theory by Nica and Speicher. The latter author gave a fantastic course on the subject. The course website contains useful information and the lectures were recorded (audio quality is not great). For the finite free probability part we will cover original papers (see this video lecture by Spielman or that one by Marcus to get a sense for this part of the course).