Three lectures on free probability

Summary

with illustrations by Michael LaCroix

These are notes from a three-lecture mini-course on free probability given at MSRI in the Fall of 2010 and repeated a year later at Harvard. The lectures were aimed at mathematicians and mathematical physicists working in combinatorics, probability, and random matrix theory. The first lecture was a staged rediscovery of free independence from first principles, the second dealt with the additive calculus of free random variables, and the third focused on random matrix models.

Introduction

These are notes from a three-lecture mini-course on free probability given at MSRI in the Fall of 2010 and repeated a year later at Harvard. The lectures were aimed at mathematicians and mathematical physicists working in combinatorics, probability, and random matrix theory. The first lecture was a staged rediscovery of free independence from first principles, the second dealt with the additive calculus of free random variables, and the third focused on random matrix models.

Most of my knowledge of free probability was acquired through informal conversations with my thesis supervisor, Roland Speicher, and while he is an expert in the field the same cannot be said for me. These notes reflect my own limited understanding and are no substitute for complete and rigorous treatments, such as those of Voiculescu, Dykema and Nica [Voiculescu et al. 1992], Hiai and Petz [2000], and Nica and Speicher [2006]. In addition to these sources, the expository articles of Biane [2002], Shlyakhtenko [2005] and Tao [2010] are very informative.

I would like to thank the organizers of the MSRI semester “Random Matrix Theory, Interacting Particle Systems and Integrable Systems” for the opportunity to participate as a postdoctoral fellow. Special thanks are owed to Peter Forrester for coordinating the corresponding MSRI book series volume in which these notes appear. I am also grateful to the participants of the Harvard random matrices seminar for their insightful comments and questions.

I am indebted to Michael LaCroix for making the illustrations which accompany these notes.

1. Lecture one: discovering the free world

1.1. Counting connected graphs. Let m_n denote the number of simple, undirected graphs on the vertex set [n] = {1,..., n}. We have m_n = 2(ⁿ₂), since each pair of vertices is either connected by an edge or not. A more subtle quantity is the number c_n of connected graphs on [n].