FREE PROBABILITY FOR PROBABILISTS

Philippe Biane

Abstract. This is an introduction to some of the most probabilistic aspects of free probability theory.

Introduction

Free probability is a non-commutative probability theory, in which the concept of independence of classical probability is replaced by that of freeness. This new concept incorporates both the probabilistic idea of non-correlation involved in the independence of random variables, and the algebraic notion of abscence of relations, like e.g. between the generators of a (non-abelian) free group. It was introduced abstractly by D. Voiculescu around 1983, in order to study some problems in the theory of von Neumann algebras, but some years later, around 1990, he realized that a concrete probabilistic model of free random variables is afforded by large independent random matrices. While this discovery has lead to an impressive series of advances in von Neumann algebras, some long standing open problems being solved with these new ideas, we shall not discuss these applications here, rather the purpose of these lectures is to try to explain to a probabilist audience why the theory of free random variables is an interesting and beautiful subject in itself, with deep connections with classical probability, and other related areas, such as harmonic analysis, and combinatorics. We hope that this brief introduction will help the reader find its way into the litterature on free probability. We shall take as a departure point a natural problem about hermitian matrices, whose solution involves free probability theory.

2. Large matrices

A frequent question, occuring both in mathematics and physics, is to determine the spectrum of the sum of two hermitian matrices. Knowing the respective spectra of two N ×N hermitian matrices A and B, the set of possible spectra for A+B is a subset of ℝ^{N}, which can be described explicitly and depends in a complicated way on the spectra of A and B. This problem is not an easy one, and indeed its solution was obtained only quite recently (see e.g. [F] for a discussion). When N becomes large, however, a remarkable phenomenon occurs, and it turns out that, roughly speaking, for almost all choices of the matrices A and B, with given spectra, the spectrum of A + B is essentially the same, and can be computed explicitly, without knowing the detailed structure of the matrices A and B (i.e. their eigenvectors). In order to give a rigourous mathematical statement corresponding to the above assertions, we shall first introduce a probability measure on the set of matrices

Typeset by AMS-TEX

having the same spectrum as A. By the spectral theorem this set consists of all matrices UAU^{∗}, where U describes the group of unitary N × N matrices. It is the orbit O_{A} of A under the adjoint action of U(N), and as such it carries a unique invariant probability measure ρ_{A}, the image of the Haar measure on U(N) by the map U ↦ UAU^{∗}.

Given an hermitian matrix A, with spectrum λ_{1}, …, λ_{N} (counted with multiplicities), we denote by ν_{A} the empirical distribution on the set of its eigenvalues, i.e. the probability measure

ν_{A} = \frac{1}{N} ∑_{j=1}^{N} δ_{λ_{j}}.

Knowledge of the spectrum (with multiplicities) and of ν_{A} are equivalent. We can now state the result we had in mind.

Theorem 1. Let for each N positive integer, A_{N} and B_{N} be two hermitian matrices, whose norm is bounded uniformly in N, and let ν_{1} and ν_{2} be two probability measures with compact supports on ℝ, such that ν_{A_{N}} → ν_{1} and ν_{B_{N}} → ν_{2} weakly as N → ∞, then there exists a probability measure, depending only on ν_{1} and ν_{2}, denoted ν_{1} ⊞ ν_{2}, such that ν_{A′_{N} +B′_{N}} → ν_{1} ⊞ ν_{2} weakly, in probability, as N → ∞, where A′_{N} and B′_{N} are random matrices chosen independently, with respective distributions ρ_{A_{N}} and ρ_{B_{N}}.

Thus, if we are given two N × N hermitian matrices (with N large), and we only know their spectra, then we can bet, with a good chance to win, that the measure ν_{A+B} is well approximated by the measure ν_{A} ⊞ ν_{B}. In other words, loosely speaking, we can compute the spectrum of A + B, knowing only the spectra of A and B!

Since any probability measure with compact support can be approximated by measures like ν_{A} for arbitrarily large matrices, Theorem 1 introduces a binary operation ⊞ on the set of compactly supported measures on the real line which, for reasons to be explained below, we shall call the free convolution of measures. This binary operation is clearly associative and commutative.

It is instructive to compare the preceding result with the case where all considered matrices are supposed to be diagonal. Let A be a self-adjoint diagonal matrix, then the set of all diagonal matrices having the same spectrum as A, the analogue of the set O_{A}, is an orbit of the symmetric group S_{N}, acting by permutation of diagonal entries. Let us denote the normalized counting measure on this set by ξ_{A}, then one has, denoting by μ ∗ ν the usual convolution of two probability measures on ℝ,

Theorem 2. Let for each N, A_{N} and B_{N} be two real diagonal matrices, and let ν_{1} and ν_{2} be two probability measures with compact supports, such that ν_{A_{N}} → ν_{1} and ν_{B_{N}} → ν_{2} weakly as N → ∞, then ν_{A′_{N} +B′_{N}} → ν_{1} ∗ ν_{2} weakly, in probability, as N → ∞, where A′_{N} and B′_{N} are random matrices chosen independently, with respective distributions ξ_{A_{N}} and ξ_{B_{N}}.

Although we do not know an adequate reference, this last result is probably well known. In fact, it is not difficult to deduce it from known concentration of measure results on the symmetric group, as e.g. in [M].

Coming back to free convolution, note that we have not said how to compute explicitly the measure ν_{1} ⊞ ν_{2} in terms of ν_{1} and ν_{2}. This is where free probability comes in. As we shall see in the next section, Theorem 1 is the consequence of a more general result concerning large hermitian matrices, namely the fact that they give rise, asymptotically, to free random variables.

3. Free random variables

Before we describe the connection with large matrices, we introduce the necessary notions from free probability. We start with some purely algebraic definitions.

Let A be a complex algebra with a unit, and φ a ℂ-valued linear form on A, such that φ(1) = 1. Although, in the examples that we shall consider, the algebra A will always be non-commutative, it will be convenient to think of the elements of this algebra as random variables, while the map φ should be considered as the expectation map of classical probability theory (here and in the sequel, we use the word classical for usual probability theory, as opposed to the non-commutative theory we develop).

We now introduce the basic notion of free probability theory.

Definition 1. Let I be a set of indices, and B_{i}, for i ∈ I, be subalgebras of A, containing the unit, then the algebras B_{i}; i ∈ I are called free if one has φ(a_{1} … a_{n}) = 0 each time φ(a_{j}) = 0 for all j = 1, …, n and a_{j} ∈ B_{i_{j}} for some indices i_{1} ≠ i_{2} ≠ … ≠ i_{n}.

Pursuing our analogy with classical probability, if we think of the algebras B_{i} as algebras of random variables, measurable with respect to some sub-sigma field, then the above definition is a non-commutative analogue of the definition of independent subalgebras. In fact, although it might not be obvious at first sight, this definition captures the essence of both the notion of algebraic independence, and that of independence of sigma-algebras in classical probability. Let us stress however, that this definition is not a non-commutative extension of the notion of independence, indeed algebras generated by independent random variables, in the sense of classical probability theory, are not free in the sense of the above definition.

Before making some elementary comments, we shall give a convenient definition, whose analogy with classical probability should be obvious.

Definition 2. Let H_{i}; i ∈ I be subsets of A, they are called free if the subalgebras B_{i}(= unital algebra generated by H_{i}), for i ∈ I, are free.

In order to get acquainted with freeness we shall make a few computations. So let B, C ⊂ A be two free subalgebras, and b ∈ B, c ∈ C, then we can write b = b′ + b̄ where b̄ = φ(b)1, so that φ(b′) = 0. Similarly we have c = c′ + c̄. Then using the definition of freeness, we see that φ(b′c′) = 0, thus

φ(bc) = φ((b′ + b̄)(c′ + c̄))
= φ(b′c′ + b′c̄ + b̄c′ + b̄c̄)
= φ(b)φ(c)

Here we have used φ(b̄c′) = φ(φ(b)1.c′) = φ(b)φ(c′) = 0, and similarly φ(b′c̄) = 0.

This means that for two free elements b and c, their expectations factorize, exactly as in the case of independent variables. Let us now look at some more subtle example. Take b_{1}, b_{2} ∈ B, and c_{1}, c_{2} ∈ C, then we can expand

φ(b_{1}c_{1}b_{2}c_{2}) = φ((b′_{1} + b̄_{1})(c′_{1} + c̄_{1})(b′_{2} + b̄_{2})(c′_{2} + c̄_{2}))
= φ(b′_{1}c′_{1}b′_{2}c′_{2} + b′_{1}c′_{1}b′_{2}c̄_{2} + b′_{1}c′_{1}b̄_{2}c′_{2} + b′_{1}c′_{1}b̄_{2}c̄_{2}+
b′_{1}c̄_{1}b′_{2}c′_{2} + b′_{1}c̄_{1}b′_{2}c̄_{2} + b′_{1}c̄_{1}b̄_{2}c′_{2} + b′_{1}c̄_{1}b̄_{2}c̄_{2}+
b̄_{1}c′_{1}b′_{2}c′_{2} + b̄_{1}c′_{1}b′_{2}c̄_{2} + b̄_{1}c′_{1}b̄_{2}c′_{2} + b̄_{1}c′_{1}b̄_{2}c̄_{2}+
b̄_{1}c̄_{1}b′_{2}c′_{2} + b̄_{1}c̄_{1}b′_{2}c̄_{2} + b̄_{1}c̄_{1}b̄_{2}c′_{2} + b̄_{1}c̄_{1}b̄_{2}c̄_{2})

Using the fact that φ(b′_{1}c′_{1}b′_{2}c′_{2}) = 0 by the freeness property, we are left with terms in which some expectation factorizes. For example we shall treat the term φ(b′_{1}c̄_{1}b′_{2}c′_{2}) = φ(c_{1})φ(b′_{1}b′_{2}c′_{2}). By the factorization property already obtained one has φ(b′_{1}b′_{2}c′_{2}) = φ(b′_{1}b′_{2})φ(c′_{2}) = 0. The other terms can be treated by similar considerations, and after some straightforward manipulations we arrive at the formula

φ(b_{1}c_{1}b_{2}c_{2}) = φ(b_{1}b_{2})φ(c_{1})φ(c_{2}) + φ(b_{1})φ(b_{2})φ(c_{1}c_{2}) − φ(b_{1})φ(b_{2})φ(c_{1})φ(c_{2})

We observe that the computation of the expectation of the product b_{1}c_{1}b_{2}c_{2} can be reduced to the computation of expectations in the subalgebras B and C. This also shows that the result is different from the one we would have obtained with independent (commuting) random variables.

It is not difficult to see that the above computation can be generalized. More precisely, taking care of how the terms are successively reduced, the reader should check, and it is a good exercise do so, that the following is true

Proposition 1. Let B_{i}; i ∈ I be free subalgebras in A, and a_{1}, …, a_{n} ∈ A such that for all j = 1, …, n, one has a_{j} ∈ B_{i_{j}} for some i_{j} ∈ I. Let Π be the partition of {1, …, n} determined by j ∼ k if i_{j} = i_{k}. For each partition π of {1, …, n}, let φ_{π} = ∏_{{\{j_{1},... ,jr \}∈π \atop j_{1}<...<jr}} φ(a_{j_{1}} … a_{j_{r}}), then there exists universal coefficients c(π, Π),

such that

φ(a_{1} … a_{n}) = ∑_{π≤Π} c(π, Π)φ_{π}

where the sum is over partitions π which are finer than Π.

In particular, this shows that φ(a_{1} … a_{n}) can be computed explicitly in terms of the restriction of φ to the algebras B_{i}. This is a reminiscence of the fact that the joint distribution of a family of independent random variables is completely determined if we know the distribution of each of the random variables.

As the example that we treated above suggests, the algorithm we have described for computing coefficients c(π, Π) leads quickly to intractable calculations. Finding an explicit formula for these coefficients is a non trivial combinatorial problem, which fortunately has been solved by R. Speicher. We shall come back to describe his solution later. Let us just mention for the moment that it involves a certain class of partitions, called “non-crossing”.

It is time now to state a result showing that the notion we have introduced is meaningful, i.e. that there exist non-trivial examples. In this respect the situation is optimum, and we have Theorem 3. Let B_{i}; i ∈ I be complex unital algebras, equipped with normalized (i.e. φ_{i}(1) = 1) linear forms φ_{i}, then there exists an algebra A, with a normalized linear form φ, and unital injective morphisms ι_{i} : B_{i} → A, satisfying φ_{i} = φ ∘ ι_{i}, and the algebras ι_{i}(B_{i}); i ∈ I, are free in A.

The construction of the algebra A consists in enforcing the freeness condition in a rather straightforward way. One defines it as the “free amalgamated product” of the B_{i}, more precisely, denote by B′_{i}, for all i ∈ I, the subspace of elements in B_{i} with zero expectation, and then take for A the direct sum of ℂ.1 (1 is the unit in A), and all spaces B′_{i_{1}} ⊗ … ⊗ B′_{i_{n}}, where i_{1} … i_{n} runs over all finite sequences in I satisfying i_{1} ≠ i_{2} ≠ … ≠ i_{n}. In order to turn A into an algebra, we need to specify the product of two elements (a_{1} ⊗ … ⊗ a_{n})(b_{1} ⊗ … ⊗ b_{m}). If a_{n} and b_{1} belong to distinct algebras, then the product is simply a_{1} ⊗ … ⊗ a_{n} ⊗ b_{1} ⊗ … ⊗ b_{m}, which belongs to A. If a_{n} and b_{1} belong to the same algebra B_{i}, then φ_{i}(a_{n}b_{1}) is not zero in general, so we write a_{n}b_{1} = (a_{n}b_{1})′ + \overline{a_{n}b_{1}}, and put

\begin{matrix}(a_{1} ⊗ … ⊗ a_{n})(b_{1} ⊗ … ⊗ b_{m}) = & a_{1} ⊗ … ⊗ a_{n−1} ⊗ (a_{n}b_{1})′ ⊗ b_{2} ⊗ … ⊗ b_{m} \\ + φ_{i}(a_{n}b_{1})(a_{1} ⊗ … ⊗ a_{n−1})(b_{2} ⊗ … ⊗ b_{m}) \end{matrix}

Since the “length” of (a_{1} ⊗ … ⊗ a_{n−1})(b_{2} ⊗ … ⊗ b_{m}) is m + n − 2, the computation can be repeated, and we are done after a finite number of steps. The injection ι_{i} is defined by ι_{i}(b) = φ_{i}(b).1 + b′, and the linear form φ by φ(a.1) = a for a ∈ ℂ, and φ = 0 on each space B′_{i_{1}} ⊗ … ⊗ B′_{i_{n}}. It is not difficult to check that the pair (A, φ) fulfills the required conditions.

4. Free convolution

We are now going to be more specific about the kind of algebras that we are considering. Since we want to do some probability theory, it will be convenient to be able to say that a function of a random variable is again a random variable, and that a random variable X has a distribution, namely a probability measure μ_{X}, such that φ(f(X)) = ∫ f(x)μ_{X}(dx) for any bounded function f. The cleanest way to do that is to assume that our algebra A is a von Neumann algebra, which means that it is an algebra of bounded operators in some complex Hilbert space H, closed under taking adjoints of operators, and under taking limits for the weak operator topology (i.e. simple weak convergence on H). Since we want φ to be an analogue of the expectation map with respect to a probability measure, we need some positivity assumption. Positivity here will be taken in the sense of operator theory, namely an element X of A will be positive if it is a self-adjoint positive operator on H. The positivity requirement on φ is that it takes nonnegative values on positive operators in A. We shall also assume that φ is continuous for the weak operator topology. You do not need to be familiar with von Neumann algebras in order to be able to read the following, in fact the only property of a von Neumann algebra that we will use is the stability with respect to functional calculus, namely if X ∈ A is a self-adjoint operator on H, then the operator f(X), which can be defined by functional calculus for any bounded Borel function f on ℝ, still belongs to the algebra A. Also, if X is self-adjoint then the map f ↦ φ(f(X)), defined for Borel bounded functions, is given by f ↦ ∫_{ℝ} f(x)μ_{X}(dx), for a unique probability measure μ_{X}, with compact support on ℝ. The probability measure μ_{X} is called the distribution of X. Thus, self-adjoint operators in A behave somewhat like (real valued) random variables.

If A is a von Neumann algebra, and φ is a normalized, weakly continuous, positive linear form on A, we shall call the couple (A, φ) a non-commutative probability space.

It turns out that the amalgamated free product of algebras exists in the category of non-commutative probability spaces, namely if the (B_{i}, φ_{i}) in Theorem 3 are noncommutative probability spaces, then we can choose for (A, φ) a non-commutative probability space, and the ι_{i} are weakly continuous. This can be seen by applying the GNS construction to the amalgamated free product constructed after Theorem 3. We refer to [VDN] for a detailed proof.

We are now in position to give the free-probabilistic definition of the free convolution of measures, which was introduced in Theorem 1. Let μ and ν be probability measures with compact support on ℝ, then there exists a non-commutative probability space (A, φ) and self-adjoint elements X, Y in A with respective distributions μ and ν, such that X an Y are free. In order to construct (A, φ) and X, Y, we can do the following: take the algebra L^{∞}(ℝ, μ), which can be considered as a von Neumann algebra of operators on L^{2}(μ), elements of L^{∞}(ℝ, μ) acting by multiplication on L^{2}(μ). The expectation map φ_{μ} (integration with respect to μ) turns the pair (L^{∞}(ℝ, μ), φ_{μ}) into a non-commutative probability space. The map x ↦ x on ℝ defines a self-adjoint element of L^{∞}(ℝ, μ) (since μ has compact support), which we call X. By the general construction described above, there exists a noncommutative probability space, containing (L^{∞}(ℝ, μ), φ_{μ}) and (L^{∞}(ℝ, ν), φ_{ν}) as free subalgebras, and thus we get a non-commutative probability space containing two free elements X and Y with respective distributions μ and ν. Since X and Y are bounded selfadjoint operators, the distribution of X+Y is a probability measure with compact support on ℝ, and it is characterized by its moments. Expanding the expression φ((X + Y)^{n}) and using Proposition 1, we see that the moments φ((X + Y)^{n}) can be computed as polynomial functions of the moments of μ and ν, so that the distribution of X + Y really depends only on the measures μ and ν and not on the particular realization of the operators X and Y, thus we can define μ⊞ν as the distribution of X + Y, where X and Y is any pair of free random variables, with respective distributions μ and ν. Clearly this definition makes the operation ⊞ the free analogue of the convolution of measures in classical probability.

We shall now recover Theorem 1 from the following result, which shows that large hermitian matrices give an asymptotic model for free random variables.

Theorem 4. Let A_{N}, B_{N}, ν_{1} and ν_{2} be as in the statement of Theorem 1, and let X, Y be two self-adjoint elements of some non-commutative probability space (A, φ), with respective distributions ν_{1} and ν_{2}, then for every non-commutative polynomial in two indeterminates, P, one has \frac{1}{N} tr(P (A′_{N}, B′_{N})) → φ(P (X, Y)) in probability, as N → ∞, where A′_{N} and B′_{N} are random matrices chosen independently, with respective distributions ρ_{A_{N}} and ρ_{B_{N}}.

Applying Theorem 4 to the non-commutative polynomials P_{n}(X, Y) = (X+Y)^{n}, we see that the moments of A′_{N} +B′_{N} converge in probability towards that of X +Y. Since we are dealing with compactly supported measures, we conclude to the weak convergence of the associated distributions, and we get Theorem 1.

We shall not give a proof of Theorem 4 here, since this is a far from easy result. This is a consequence of a more general result from the paper [Vo2] (see also [VDN]), and another, more direct proof can also been found in [X].

It is now time we give a formula for computing the free convolution of two measures. For convolution of probability measures, one way to perform this computation is to use the Fourier transform, which converts convolution into multiplication. Observe that for probability measures with compact support, the logarithm of the Fourier transform can be expanded into a power series

log(∫_{ℝ} e^{itx}μ(dx)) = ∑_{n=1}^{∞} σ_{n}(μ)(it)^{n}.

This follows from a substitution of the power series

∫_{ℝ} e^{itx}μ(dx) = 1 + ∑_{n=1}^{∞} \frac{(it)^{n}}{n!} ∫_{ℝ} x^{n}μ(dx)

into the expansion of log(1 + z). The coefficients σ_{n}(μ) are polynomial functions in the moments of μ, called the cumulants of the measure μ, and the multiplicativity property of the Fourier transform of convolution shows that these cumulants linearize the convolution, namely σ_{n}(μ ∗ ν) = σ_{n}(μ) + σ_{n}(ν). It turns out that the free convolution admits a similar description in terms of so-called “non-crossing cumulants”, which are defined as follows. Let

G_{μ}(ζ) = ∫_{ℝ} \frac{1}{ζ − t}dμ(t)

be the Cauchy transform of μ. This defines an analytic function on ℂ\ℝ, such that G_{μ}(ζ̄) = \overline{G_{μ}(ζ)}, and G_{μ}(ℂ^{+}) ⊂ ℂ^{−} where ℂ^{+} and ℂ^{−} denote respectively the sets of complex numbers with positive and negative imaginary part. We can expand this function in a Laurent series involving the moments of μ

G_{μ}(ζ) = ∑_{n=0}^{∞} ζ^{−n−1} ∫_{ℝ} x^{n}μ(dx)

This series can be inverted formally with respect to ζ, the formal inverse having the form

K_{μ}(z) = \frac{1}{z} + ∑_{n=1}^{∞} C_{n}z^{n−1}

where the coefficients C_{n} are polynomial functions of the moments m_{k} = ∫_{ℝ} x^{k}μ(dx) of μ. The most convenient way to proceed in order to compute the C_{n} is to write the equation defining K_{μ} as

G_{μ}(ζ)K_{μ}(G_{μ}(ζ)) = 1 + ∑_{n=1}^{∞} C_{n}G_{μ}(ζ)^{n} = ζG_{μ}(ζ)

Equating the coefficients of ζ^{−n} in the second and third member of the above equality, we can evaluate the C_{n} in terms of the moments recursively. The first few values are

\begin{aligned}C_{1} & = m_{1} \\ C_{2} & = m_{2} − m_{1}^{2} \\ C_{3} & = m_{3} − 3m_{1}m_{2} + 2m_{1}^{3} \end{aligned}

These formulas can be inverted, and we get the moments in terms of the C_{n} as

\begin{matrix}m_{1} & = C_{1} \\ m_{2} = & C_{2} + C_{1}^{2} \\ m_{3} = & C_{3} + 3C_{1}C_{2} + C_{1}^{3} \end{matrix}

The coefficients C_{n}, which we shall call the free cumulants of the measure μ play the role of cumulants for the free convolution, namely one has

Theorem 5. For all compactly supported measures μ and ν, on ℝ, and all n ≥ 1, one has

C_{n}(μ ⊞ ν) = C_{n}(μ) + C_{n}(ν).

Since one can recover the moments of a measure from its free cumulants, this determines completely the measure μ ⊞ ν.

This result was first proved by Voiculescu in [Vo1], using free creation and annihilation operators (the formula has also been discovered independently around the same time, in the more restrictive context of random walks on free products of groups, see [W]). Later, Speicher used his combinatorial approach to freeness to give another proof of Theorem 5. We shall describe Speicher’s proof in section 5.

Let us give an example of the computation of ⊞. Let

μ = ν = \frac{1}{2}(δ_{0} + δ_{1})

one has

G_{μ}(ζ) = \frac{ζ − 1/2}{ζ(ζ − 1)} and K_{μ}(z) = \frac{z + 1 + \sqrt{1 + z^{2}}}{2z}

This gives

K_{μ⊞μ}(z) = 1 + \frac{1}{z}\sqrt{1 + z^{2}}

and

G_{μ⊞μ}(ζ) = \frac{1}{\sqrt{ζ(ζ − 1)}}

The Cauchy transform can be inverted to give

μ ⊞ μ(dx) = \frac{1}{π\sqrt{x(2 − x)}}dx on [0, +2].

This is the famous arcsine distribution (here on the interval [0, +2]). If we recall Theorem 3, we have the following striking interpretation of this computation: take a large integer N, and two subspaces of ℂ^{N}, of dimension [N/2], then for most choices of these subspaces, the sum of the corresponding orthogonal projections has an eigenvalue distribution which is well approximated by the arcsine distribution.

4. Theory of addition of free random variables

Once one has defined the free convolution of measures, one may try to develop the theory of addition of free random variables in parallel with the theory of addition of independent random variables. It turns out that most of the well known classical results, such as the law of large numbers, the central limit theorem, or the Lévy-Khintchine formula, have free analogues, and this develops into a beautiful new branch of mathematics, which sheds new lights on some well known results like the Wigner theorem on spectra of random gaussian matrices. We shall give a brief survey of these results below, but before that we need to extend the free convolution to probability measures with unbounded support.

4.1 Free convolution of measures with unbounded support. Let μ be an arbitrary probability measure on ℝ and let

G_{μ}(ζ) = ∫_{ℝ} \frac{1}{ζ − t}dμ(t)

be its Cauchy transform. Since μ may very well have no moment at all, we no longer have the expansion of G_{μ} into a power series in ζ^{−1}, however the following is nevertheless true, let

Θ_{α,β} = {z = x + iy | y < 0; αy < x < −αy; |z| ≤ β}

For every α > 0, there exists a real number β > 0 such that the function G_{μ} has a right inverse defined on the domain Θ_{α,β}, taking values in some domain of the form

Γ_{γ,λ} = {z = x + iy | y > 0; −γy < x < γy; |z| ≥ λ}

with γ, λ > 0. Call K_{μ} this right inverse, and let R_{μ}(z) = K_{μ}(z) − \frac{1}{z}. The function R_{μ} is called the R-transform of the measure μ. Given another probability measure ν on ℝ, we shall define a new probability measure μ ⊞ ν by the requirement that

R_{μ⊞ν} = R_{μ} + R_{ν}

on some domain of the form Θ_{α,β}, where these three functions are defined. It turns out that this definition is meaningful, and it is clear that it coincides with the previous definition in the case where μ and ν have compact support. In fact, there is also an interpretation of μ ⊞ ν as the distribution of the sum of two free (unbounded in general) self-adjoint elements affiliated to some non-commutative probability space, see [BV] for details.

4.2 Law of large numbers. The most general formulation of the law of large numbers for sums of independent equidistributed random variables asserts that the distribution of \frac{1}{n}(X_{1} +…+X_{n} −M_{n}) converges weakly towards the Dirac measure at zero, for some constants M_{n}, if and only if the common distribution μ of the X_{j} satisfies tμ(ℝ \ [−t, t]) → 0 as t → ∞. It turns out that exactly the same result holds if one replaces independent random variables by free ones. This is proved in [BP].

4.3 The Central Limit Theorem. Let μ be a probability measure having zero mean and finite variance σ, then one has R_{μ}(z) ∼ σz as z → 0. Let now X_{1}, …, X_{n}, … be a sequence of identically distributed free random variables, with common distribution μ, then it is easy to see that the distribution of \frac{X_{1}+...X_{n}}{\sqrt{n}} has R-transform \sqrt{n}R_{μ}(\frac{z}{\sqrt{n}}), which converges, as n → ∞, towards σz. Using continuity properties of the R-tranform, it is not difficult to see that this implies the following free central limit theorem

Theorem 6. The distribution of \frac{X_{1}+...+X_{n}}{\sqrt{n}} converges weakly, as n → ∞, towards the distribution with R-transform σz.

The distribution appearing in the free central limit theorem can be computed, by inverting the R-transform, it is the famous Wigner semi-circular distribution given by the density \frac{1}{2πσ} \sqrt{4σ − x^{2}} on the interval [−2\sqrt{σ}, 2\sqrt{σ}]. This central limit theorem, together with Theorem 3 on asymptotic freeness of random matrices, provides a conceptual framework for the well known result of Wigner, on the asymptotic behaviour of the spectra of large random matrices with gaussian entries.

4.4 Infinitely divisible distributions. There is a notion of infinitely divisible measures for the free additive convolution, which is the obvious one, namely a probability measure μ on the real line is said to be freely infinitely divisible if for every positive integer n, there exists a measure μ_{n} such that μ_{n}^{⊞n} = μ. The following characterization of freely infinitely divisible measures on ℝ has been obtained by Bercovici and Voiculescu [BV].

Theorem 7. A probability measure μ, on ℝ is infinitely divisible if and only if the function R_{μ} has an analytic continuation to the whole of ℂ^{+}, with values in ℂ^{−} ∪ℝ, and one has

lim_{y→0,y∈ℝ} yR_{μ}(iy) = 0

Furthermore any analytic function R having the above properties is the R-transform of some probability measure.

Functions such as those appearing in Theorem 7, have a Nevanlinna representation, which can be seen here as the free analogue of the Lévy-Khintchine formula. More precisely, let R be the R-transform of some infinitely divisible probability measure, then there exists a real number α, and a finite positive measure ν, on ℝ, such that

R(z) = α + ∫_{−∞}^{+∞} \frac{z + t}{1 − tz} dν(t)

The extreme points in this integral representation have an interpretation similar to the one of the classical Lévy-Khintchine formula, namely the function R(z) = α is the R-transform of a Dirac mass at the point α, the function R(z) = σz, corresponding to a point mass at zero for ν, is the R transform of the semi-circular distribution with variance σ. Finally the probability measure with R-transform R(z) = λ\frac{z+t}{1−tz} is the free analogue of the Poisson distribution, namely, it is the weak limit, as n → ∞ of the measures ((1 − \frac{λ}{n})δ_{0} + \frac{λ}{n}δ_{t})^{⊞n} (the “free binomial distributions”).

4.5 Stable distributions. One can define stable distributions exactly as for classical convolution, namely, a probability measure on ℝ is called stable, if and only if the set of probability measures obtained from μ by applying affine transformations of ℝ, is stable under free convolution. The set of all free stable probability measures on the real line has been determined by Bercovici and Voiculescu [BV]. It turns out that there is a natural one-to-one correspondance between stable distributions and free stable distributions, and the domains of attraction are the same in the classical and the free cases. In fact, any free stable distribution is the image by an affine transformation of a distribution whose R-transform belongs to the following list

(1) R(z) = e^{iπθ}z^{α−1} where 1 < α ≤ 2, and (α − 2) ≤ θ ≤ 0.

(2) R(z) = a + b log z where a ∈ ℂ^{+} ∪ ℝ and b ≥ −ℑa/π.

(3) R(z) = e^{iπθ}z^{α−1} where 0 < α < 1, and 1 ≤ θ ≤ 1 + α. In particular, the Cauchy distribution is a free stable distribution of stability index one.

For these results, see [BVB]

5. Speicher’s combinatorial approach to freenes

We shall now describe Speicher’s combinatorial approach to the computation of the coefficients c(π, Π) of Proposition 1, and some applications. A thorough discussion is given in [Sp2].

Let S be a totally ordered set. A partition of the set S is said to have a crossing if there exists a quadruple (i, j, k, l) ∈ S^{4}, with i < j < k < l, such that i and k belong to some class of the partition and j and l belong to another class. If a partition has no crossing, it is called non-crossing. The set of all non-crossing partitions of S is denoted by NC(S), it is a lattice with respect to the dual refinement order (such that π ≤ σ if π is a finer partition than σ).

When S = {1, …, n}, with its natural order, we will use the notation NC(n). Here is an example with n = 8, π = {{1, 4, 5}, {2}, {3}, {6, 8}, {7}}.

fig.1

We draw a segment joining each point with the next point in the same class of the partition. The non-crossing condition means that segments should not intesect inside the circle.

Let (A, φ) be a non-commutative probability space, then we shall define a family R^{(n)} of n-multilinear forms on A, for n ≥ 1, by the following formula

φ(a_{1} … a_{n}) = ∑_{π∈NC(n)} R[π](a_{1}, …, a_{n})

Here, for π ∈ NC(n), one has

R[π](a_{1}, …, a_{n}) = ∏_{V ∈π} R^{(|V |)}(a_{V})

where a_{V} = (a_{j_{1}}, …, a_{j_{k}}) if V = {j_{1}, …, j_{k}} is a class of the partition π, with j_{1} < j_{2} < … < j_{k}. In particular R[1_{n}] = R^{(n)} if 1_{n} is the partition with only one class. For example, one has, for all a ∈ A,

φ(a) = R(a)

(we forget the superscript (n) when it is clear which n is considered), for n = 2

φ(a_{1}a_{2}) = R(a_{1}, a_{2}) + R(a_{1})R(a_{2})

thus

R(a_{1}, a_{2}) = φ(a_{1}a_{2}) − φ(a_{1})φ(a_{2})

is the covariance of a_{1} and a_{2}. Finally,

\begin{matrix}φ(a_{1}a_{2}a_{3}) = & R(a_{1}, a_{2}, a_{3}) + R(a_{1})R(a_{2}, a_{3}) + R(a_{2})R(a_{1}, a_{3}) \\ + R(a_{3})R(a_{1}, a_{2}) + R(a_{1})R(a_{2})R(a_{3}) \end{matrix}

which yields

\begin{matrix}R(a_{1}, a_{2}, a_{3}) = & φ(a_{1}a_{2}a_{3}) − φ(a_{1})φ(a_{2}a_{3}) − φ(a_{2})φ(a_{1}a_{3}) \\ − φ(a_{3})φ(a_{1}a_{2}) + 2φ(a_{1})φ(a_{2})φ(a_{3}) \end{matrix}

In general, for each n, one has

φ(a_{1} … a_{n}) = R^{(n)}(a_{1}, …, a_{n}) + terms involving R^{(k)} for k < n

so that the R^{(n)} are well defined and can be computed by induction on n.

The non-crossing cumulants can be expressed explicitly in terms of the moments by the following formula

R^{(n)}(a_{1}, …, a_{n}) = ∑_{π∈NC(n)} Moeb(π)φ[π](a_{1}, …, a_{n})

Here φ[π](a_{1}, …, a_{n}) = ∏_{V ∈π} φ(a_{j_{1}} … a_{j_{k}}) where V = {j_{1}, …, j_{k}} are the classes of π, and Moeb is the Möbius function on NC(n) defined by

Moeb(π) = ∏_{V ∈π}(−1)^{|V |}c_{|V |−1}

where c_{n} = \frac{(2n)!}{n!(n+1)!} is the n^{th} Catalan number.

The connection between non-crossing cumulants and freeness is the following result of section 4 of [Sp1].

Proposition 2. Let (B_{i})_{i∈I} be free subalgebras of A, and a_{1}, …, a_{n} ∈ A be such that a_{j} belongs to some B_{i_{j}} for each j ∈ {1, 2, …, n}. Then R(a_{1}, …, a_{n}) = 0 if there exists some j and k with i_{j} ≠ i_{k}.

Proposition 2 is the key to the computation of the coefficients c(π, Π) of Proposition 1. Indeed, let a_{1}, …, a_{n} be an arbitrary sequence in A, such that each a_{j} belongs to one of the algebras B_{i}, then we have

φ(a_{1} … a_{n}) = ∑_{π∈NC(n)} R[π](a_{1}, …, a_{n})

and in this sum the terms corresponding to partitions π having a class containing two elements j, k such that a_{j} and a_{k} belong to distinct algebras give a zero contribution. Thus we have to sum over partitions in which all j belonging to a certain class are such that a_{j} belongs to the same algebra, and the value of R[π](a_{1}, …, a_{n}) can be expanded in terms of the restriction of φ to each of the subalgebras.

The multilinear forms R^{(n)} allow us to recover the free cumulants, indeed one has

∫_{ℝ} x^{n} μ(dx) = φ(X^{n}) = ∑_{π∈NC(n)} R[π](X, …, X)

Proposition 3. Let X ∈ A be selfadjoint and have distribution μ(dx), then the free cumulants of the measure μ are given by the formula C_{n}(μ) = R^{(n)}(X, …, X), for n = 1, 2 . . ..

Using Propositions 2 and 3, we can now give a proof of Theorem 5. Let X and Y be free random variables with respective distributions μ and ν, then the cumulants of μ ⊞ ν are given, according to Proposition 3, by

C_{n}(μ ⊞ ν) = R^{(n)}(X + Y, …, X + Y)

Since R^{(n)} is an n-linear form, we can expand R^{(n)}(X +Y, …, X +Y), into a sum of terms R^{(n)}(Z_{1}, Z_{2}, …, Z_{n}), where each Z_{i} is either X or Y. Applying Proposition 2, we see that all these terms are zero, except R^{(n)}(X, …, X) and R^{(n)}(Y, …, Y), thus we have

R^{(n)}(X + Y, …, X + Y) = R^{(n)}(X, …, X) + R^{(n)}(Y, …, Y)

and Theorem 5 follows from Proposition 3 again. The proofs of Propositions 2 and 3 can be found in [Sp1] or [Sp2].

6. Some further topics

6.1 Multiplicative free convolution. Given two unitary elements U, V, which are free in some non-commutative probability space (A, φ), we can form their product, which is again a unitary element. The distributions of U and V are this time probability measures say μ and ν, on the set T of complex numbers of modulus one, and the distribution of UV, which depends only on μ and ν, can be computed by the so-called Σ-transform. Let us introduce the ψ-function of a probability measure μ, on T, by

ψ_{μ}(z) = ∫_{T} \frac{zξ}{1 − zξ} dμ(ξ)

This is a convergent power series in D = {z ∈ ℂ | |z| < 1}, the open unit disk of ℂ, such that ψ_{μ}(0) = 0. Let M_{∗} be the set of probability measures on T such that ∫_{T} ξdμ(ξ) ≠ 0. If μ ∈ M_{∗}, the function \frac{ψ_{μ}}{1+ψ_{μ}} has a right inverse, called χ̃_{μ}, defined in a neighbourhood of 0, such that χ̃_{μ}(0) = 0, and we let Σ_{μ}(z) = \frac{1}{z} χ̃_{μ}(z) be the Σ-transform of μ. Then, for any measures μ, ν ∈ M_{∗}, one has μ ⊠ ν ∈ M_{∗} and

Σ_{μ⊠ν}(z) = Σ_{μ}(z)Σ_{ν}(z)

in some neighbourhood of zero where these three functions are defined.

This formula was first found by Voiculescu in [V2]. His proof was quite complicated, and a simpler one has been given by U. Haagerup. A proof using non-crossing partitions, due to Nica and Speicher is in [NS1].

There is an analogue, for free multiplicative convolution on T, of the Lévy-Khinchine formula. This states that a probability measure on T is infinitely divisible, for the free multiplicative convolution, if and only if its Σ transform can be written as Σ_{μ}(z) = exp(u(z)) where u is an analytic function on D, taking values with nonnegative real parts. Such a function has a representation of the form

u(z) = iα + ∫_{T} \frac{1 + ζz}{1 − ζz} dν(ζ)

for some finite positive measure ν on T, and real number α.

Finally one can also define multiplicative free convolution for measures on ℝ_{+}. We shall refer to [BV] for these topics.

We shall here make a remark on some features which distinguish free probability from classical probability. Let μ be a probability measure on ℝ, and let G be (a suitable branch of) the logarithm of its Fourier transform, then the set of positive real numbers t such that tG is again the logarithm of the Fourier transform of some probability measure is a closed additive subsemigroup of ℝ_{+}, containing the positive integers. It is equal to ℝ_{+}, exactly when μ is infinitely divisible, but it can also be reduced to the set of positive integers (e.g if μ is a Bernoulli distribution). In free probability, the analogue of the function G is the R-transform. Again, the set of t such that tR_{μ} is the R-transform of some probability measure is a closed additive subsemigroup of ℝ_{+}, but this time this subsemigroup always contains the interval [1, +∞[. In fact the measure with R-transform tR_{μ} has a nice description in terms a free compression. Namely, let X be self-adjoint, with distribution μ in (A, φ), and let π be a self-adjoint projection in A, free with A, and such that φ(π) = \frac{1}{t}, with t ∈ [1, ∞[. The distribution of μ is the Bernoulli distribution (1 − \frac{1}{t})δ_{0} + \frac{1}{t}δ_{1}. The set πAπ of elements of the form πZπ, for Z ∈ A, is an algebra, with π as a unit, and (πAπ, tφ) is a non-commutative probability space. The distribution of the element tπXπ in (πAπ, tφ), has a distribution whose R-transform is given by tR_{μ} (try to show this using what you know about R-transforms and non-crossing partitions!). This shows that tR_{μ} is the R-transform of some probability measure, for all t ≥ 1. As the example of Bernoulli distribution shows, there are probability measures μ for which tR_{μ} is the R-transform of some probability measure only for t ∈ {0} ∪ [1, ∞[.

In order to close this section, let us note that the problem of finding the distribution of the commutator of two free self-adjoint random variables has been solved recently by Nica and Speicher [NS2].

6.2 More about random matrices. As we have seen in section 2, free probability provides us with a good understanding of the way that spectra of large matrices behave under addition. So far we have not said anything about eigenvectors. It turns out that free probability again has something to tell us about this. We shall again consider two large hermitian matrices A and B, whose spectra are known. Thinking of A + B as a perturbation of the matrix A, we would like to know how the eigenvectors of A + B are related to those of A. It is clear that if B is small compared to A, then the eigenvectors of A + B should be close to those of A. Let us denote by γ_{1}, …, γ_{N} the normalized eigenvectors of A, associated with the eigenvalues λ_{1}, …, λ_{N}, whereas we denote by ξ_{1}, …, ξ_{N} the eigenvectors of A+B, associated with the eigenvalue ν_{1}, …, ν_{N}. The passage from the old basis to the new basis is given by the transition matrix ⟨ξ_{k}, γ_{l}⟩. In fact, since the eigenvectors are only defined up to some complex number of modulus one, we shall only consider the numbers |⟨ξ_{k}, γ_{l}⟩|^{2}, which form a bistochastic matrix. These numbers are quite hard to tackle, and they may not have a definite asymptotic behaviour, as N → ∞, so we shall evaluate them again some test functions. Let f and g be smooth functions on ℝ, then we shall look at the asymptotic behaviour of the expression

∑_{1≤k,l≤N} g(ν_{k})f(λ_{l})|⟨ξ_{k}, γ_{l}⟩|^{2}

We can rewrite this as

tr(g(A + B)f(A))

then using our result on asymptotic behaviour of large matrices, it is easy to see that, if the empirical distributions on the eigenvalues of A and B converge, as N → ∞, towards μ and ν, and we choose matrices at random as in Theorem 3, then the expression

\frac{1}{N} tr(g(A′_{N} + B′_{N})f(A′_{N}))

will converge, in probability, as N → ∞, towards

φ(g(X + Y)f(X))

where X and Y are free self-adjoint elements, with respective distributions ν_{1} and ν_{2}. In principle, we can compute the value of such an expression, for example if f and g are polynomials. In fact it is easy, by a positivity argument, to see that this value is given by ∫ f(x)g(u)ρ(dx, du) where ρ is some probability measure on ℝ^{2}. It turns out that this probability measure can be desintegrated along the values of x as ρ(dx, du) = k(x, du)μ(dx), where k(x, du) is a Markov transition kernel, which could be thought of as the limit of the bistochastic matrices |⟨ξ_{k}, γ_{l}⟩|^{2}, and this Markov kernel can be explicitly computed, namely it is characterized by

∫_{ℝ}(ζ − u)^{−1}k(x, du) = (F (ζ) − x)^{−1} for all ζ ∈ ℂ \ ℝ

for some function F on ℂ \ ℝ such that F (ζ̄) = \overline{F (ζ)}, F (ℂ^{+}) ⊂ ℂ^{+}, Im(F (ζ)) ≥ Im(ζ) for ζ ∈ ℂ^{+}, and \frac{F (iy)}{iy} → 1 as y → +∞, y ∈ ℝ, and

G_{μ}(F (ζ)) = G_{μ⊞ν}(ζ)

for all ζ ∈ ℂ \ ℝ. The map F is uniquely determined by these properties.

This result appears in [Bi], where it is applied to the theory of processes with free increments.

References

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[BP] H. Bercovici and V. Pata, The law of large numbers for free identically distributeds random variables, Ann. Proba. 24 (1996), 453–465.

[BPB] H. Bercovici and V. Pata, Domains of attraction of free stable distributions; with an appendix by P. Biane on the density of free stable distributions, Ann. Math. (to appear).

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