## What is Free Probability?

### A motivating example

Consider the following three natural questions from the fields of operator algebras,random matrices, and combinatorics, respectively.

- What is the distribution of the real part of the one-sided shift with respect

to the canonical vacuum expectation state?

- What is the asymptotic eigenvalue distribution of a Gaussian random matrix

in the limit of large matrix size?

- In how many ways can one connect 2
*n*points on a circle pairwisely by*n*

non-intersecting chords?

and 2 yield the semicircular distribution, and the answer to question 3 is given by the

Catalan numbers which are actually the even moments of the semicircular distribution.

This coincidence is not just superficial - a conceptual reason is given by Voiculescu's

free probability theory and its basic concept of freeness.

### Origins and aspects of free probability

Free probability theory originated from questions about the structure of von Neumannalgebras related to free groups, namely those related to constructions around free products

of groups. One of the big unsolved problems in operator algebra is the question, whether

the free group factors are all isomorphic or not. Free probability theory is a very promising

attempt to attack this problem.

But also, free probability theory has evolved into a field with links to many other

quite unrelated fields: most notably random matrices, but also combinatorics, representation

theory of large groups, mathematical physics, or as applied fields as financial correlations

and wireless communications. Roland Speicher developed a combinatorial approach to free

probability theory - which explains why the third of the above questions is related to the

other two - and this has played a major role in many of those investigations.

### Non-commutative distributions in free probability

Free probability theory deals with distributions of variables. Whereas in the above mentionedexample of the semicircular distribution one has only one variable, the main interest is actually

the multivariate situation, where one has several variables. Typically those variables do not

commute, so one wants to understand

*non-commutative distributions*. Such distributions may

arise in the large

*N*limit of random multi-matrix models, as the distribution of operators in

operator algebras, or they might encode interesting combinatorial information. In each of those

fields the understanding of properties of the non-commutative distribution is related to

fundamental problems.

- In random matrix theory: to the existence and properties of the limit of multi-matrix

models; - In operator algebras: to one of the most famous open problems, namely the isomorphism

problem for free group factors; and more generally, to invariants for von Neumann algebras; - In combinatorics: to the counting of planar graphs or maps.

very powerful classical analytical tools (like Fourier transform) for dealing with those objects. In

the non-commutative, however, all those classical tools break down and one is in need of a new

kind of non-commutative analysis (aka

*free analysis*). Such a theory is still in its infancy.

In the projects of the ERC Advanced Grant NCDFP, held by Roland Speicher, the aim is to develop

the tools for dealing with such non-commutative distributions.

### Links to web pages concerning free probability

Workshop 2004 on free probability theory, Banff (including Final Report)Workshop 2008 on free probability, extensions and applications, Banff (including Final Report)

Workshop on free probability theory 2005, Oberwolfach (including Final Report)

Videos from the Focus Program on Noncommutative Distributions in Free Probability Theory

at the Fields Institute, Toronto, Canada (talks and expository lectures):

First Workshop, Inter-Program Workshop, Second Workshop

Survey articles by Roland Speicher

What is a free cumulant? (by Jonathan Novak and Piotr Sniady)

Voiculescu receives NAS award in mathematics

Conferences concerning free probability theory

Victor Perez Abreu's homepage about free probability (in Spanish)

Free probability theory on wikipedia (English)

Freie Wahrscheinlichkeitstheorie auf Wikipedia (German)

updated: 30 September 2014 Moritz Weber | Impressum |