A convenient fact which was observed and then repeatedly used in the preceding lectures is that one can construct a family of noncommutative random variables with a prescribed joint distribution (where quite often the joint distribution is indicated in the guise of a prescribed joint R-transform). Sometimes there may be more than one way of doing such a construction; an example of such a situation is the one of free semicircular families, which can be obtained by an abstract free product construction, but can also be “concretely” put into evidence by using operators of creation and annihilation on the full Fock space (cf. Lecture 7).

Of course, all the different methods for constructing a family of elements with a prescribed joint distribution are ultimately equivalent, in the respect that the calculations with moments and with cumulants performed on the family give the same results, no matter how the family was constructed. Nevertheless, there can be a substantial difference in the transparency of the calculations – it may happen that the solution to the problem we are trying to solve shows up more easily if one method of construction is used over another.

So this is, in a nut-shell, the idea of “modeling”: find a good way of constructing a family of non-commutative random variables with a given joint distribution, so that we are at an advantage when computing moments and cumulants of that family.

In the final two lectures we want to treat one of the most important and inspiring realizations of free independence. Canonical examples for free random variables appeared in the context of group algebras of free products of groups and in the context of creation and annihilation operators on full Fock spaces. These are two (closely related) examples where the occurrence of free independence is not very surprising, because its definition was just modeled according to the situation on the group (or von Neumann) algebra of the free group.

But there are objects from a quite different mathematical universe which are also free (at least asymptotically), namely special random matrices. A priori, random matrices have nothing to do with free independence and this surprising connection is one of the key results in free probability theory. It establishes links between quite different fields.

We will present in this and the next lecture the fundamental results of Voiculescu on the asymptotic free independence of special random matrices. Our approach will be quite combinatorial and fits well with our combinatorial description of free independence. In a sense, we will show that the combinatorics of free probability theory arises as the limit N → ∞ of the combinatorics of the considered N × N random matrices.

Moments of Gaussian random variables

Random matrices are matrices whose entries are classical random variables, and the most important class of random matrices are the socalled Gaussian random matrices whose entries form a Gaussian family of classical random variables.

Our main concern in this lecture will be the understanding and effective description of the sum of freely independent random variables. How can we calculate the distribution of a + b if a and b are free and if we know the distribution of a and the distribution of b. Of particular interest is the case of selfadjoint random variables x and y in a C*-probability space. In this case their distributions can be identified with probability measures on ℝ and thus taking the sum of free random variables gives rise to a binary operation on probability measures on ℝ. We will call this operation “free convolution,” in analogy with the usual concept of convolution of probability measures which corresponds to taking the sum of classically independent random variables. Our combinatorial approach to free probability theory, resting on the notion of free cumulants, will give us very easy access to the main results of Voiculescu on this free convolution via the so-called “R-transform.”

Free convolution

Definition 12.1. Let μ and ν be probability measures on ℝ with compact support. Let x and y be selfadjoint random variables in some C*-probability space such that x has distribution μ, y has distribution ν, and such that x and y are freely independent. Then the distribution of the sum x + y is called the free convolution of μ and μ and is denoted by μ ⊞ ν.

The original use of the R-transform was in connection to the problem of describing the distribution of a sum of free random variables (via the formula Ra+b = Ra + Rb, which always holds when a is free from b in some non-commutative probability space – cf. Lectures 12 and 16). Similarly, the S-transform was introduced to solve the problem of multiplication of free random variables (cf. Lecture 18). By following these lines, it is natural to ask what happens when one considers the commutator ab−ba, or the anti-commutator ab+ba of two free elements. Some remarks about this have already been made in Lecture 15. In the present lecture we will continue the discussion started there, by using the convenient language of the operation of boxed convolution.

The problem of the free commutator can be treated on two levels, which will be discussed separately.

First there is a level where one considers even random variables. At this level the problem can be solved as an application of the results on R-diagonal elements. One comes to a formula which is at the same time valid for the anti-commutator (of two free, even random variables), and which was already presented in Theorem 15.20.

Then there is the general level, where the assumption that the random variables are even is dropped. Quite surprisingly, it turns out that the free commutator (unlike the free anti-commutator) is still described in this case by the same formula as we had in the even case.

Free probability theory is a quite recent theory, bringing together many different fields of mathematics, for example operator algebras, random matrices, combinatorics, or representation theory of symmetric groups. So it has a lot to offer to various mathematical communities, and interest in free probability has steadily increased in recent years.

However, this diversity of the field also has the consequence that it is considered hard to access for a beginner. Most of the literature on free probability consists of a mixture of operator algebraic and probabilistic notions and arguments, interwoven with random matrices and combinatorics.

Whereas more advanced operator algebraic or probabilistic expertise might indeed be necessary for a deeper appreciation of special applications in the respective fields, the basic core of the theory, however, can be mostly freed from this and it is possible to give a fairly elementary introduction to the main notions, ideas and problems of free probability theory. The present lectures are intended to provide such an introduction.

Our main emphasis will be on the combinatorial side of free probability. Even when stripped from analytical structure, the main features of free independence are still present; moreover, even on this more combinatorial level it is important to organize all relevant information about the considered variables in the right way. Anyone who has tried to perform computations of joint distributions for non-commuting variables will probably agree that they tend to be horribly messy if done in a naive way.

Another important random matrix ensemble is given by Haar unitary random matrices – these are unitary matrices equipped with the Haar measure as corresponding probability measure. We will see that one can get asymptotic freeness results for Haar unitary random matrices similar to those we derived for Gaussian random matrices in the last lecture. We will also see that we have asymptotic freeness between constant matrices which are randomly rotated by a Haar unitary random matrix. (This will follow from the fact that conjugation by a free Haar unitary can be used to make general random variables free.)

Our calculations for the unitary random matrices will be of a similar kind to those from the last lecture. The main ingredient is a Wick type formula for correlations of the entries of the Haar unitary random matrices.

Haar unitary random matrices

Remark 23.1. A fundamental fact in abstract harmonic analysis is that any compact group has an analog of the Lebesgue measure, the so-called Haar measure, which is characterized by the fact that it is invariant under translations by group elements. This Haar measure is finite and unique up to multiplication with a constant, thus we can normalize it to a probability measure – the unique Haar probability measure on the compact group.

One of the main ideas in free probability theory is to consider the notion of free independence in analogy with the notion of classical or tensor independence. In this spirit, the first investigations of Voiculescu in free probability theory focused on free analogs of some of the most fundamental statements from classical probability theory. In particular, he proved a free analog of a central limit theorem and introduced and described a free analog of “convolution.” His investigations were quite analytical and centered around the concept of the “R-transform,” an analytic function which plays the same role in free probability theory as the logarithm of the Fourier transform in classical probability theory. However, in this analytic approach it is not so obvious why the R-transform and the logarithm of the Fourier transform should be analogous.

Our approach to free probability theory is much more combinatorial in nature and will reveal in a clearer way the parallelism between classical and free probability theory.

In order to see what kind of combinatorial objects are relevant for free probability theory, we will begin by giving an algebraic proof of the free central limit theorem. This approach will show the similar nature of classical and free probability theory very clearly, because the same kind of proof can be given for the classical central limit theorem. Most of the arguments will be the same, only in the very end one has to distinguish whether one is in the classical or in the free situation.