%&amstex \input amsppt.sty \magnification=1200 \hcorrection{.25in} \vcorrection{-.35in} \topmatter \title Free probability for probabilists \endtitle \author Philippe Biane \endauthor \address CNRS, DMI, \' Ecole Normale Sup\'erieure\endgraf 45, rue d'Ulm, 75005 Paris FRANCE \endaddress % Math Subject Classifications %\subjclass Primary ; Secondary \endsubjclass \abstract This is an introduction to some of the most probabilistic aspects of free probability theory. \endabstract \endtopmatter \document \head{ Introduction }\endhead 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. \head{2. Large matrices}\endhead 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 $N\times N$ hermitian matrices $A$ and $B$, the set of possible spectra for $A+B$ is a subset of $\Bbb R^N$, 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. \cite{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 having the same spectrum as $A$. By the spectral theorem this set consists of all matrices $UAU^*$, where $U$ describes the group of unitary $N\times N$ matrices. It is the orbit $\Cal O_A$ of $A$ under the adjoint action of $U(N)$, and as such it carries a unique invariant probability measure $\rho_A$, the image of the Haar measure on $U(N)$ by the map $U\mapsto UAU^*$. Given an hermitian matrix $A$, with spectrum $\lambda_1,\ldots,\lambda_N$ (counted with multiplicities), we denote by $\nu_A$ the empirical distribution on the set of its eigenvalues, i.e. the probability measure $$\nu_A={1\over N}\sum_{j=1}^N\delta_{\lambda_j}.$$ Knowledge of the spectrum (with multiplicities) and of $\nu_A$ are equivalent. We can now state the result we had in mind. \proclaim{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 $\nu_1$ and $\nu_2$ be two probability measures with compact supports on $\Bbb R$, such that $\nu_{A_N}\to\nu_1$ and $\nu_{B_N}\to\nu_2$ weakly as $N\to\infty$, then there exists a probability measure, depending only on $\nu_1$ and $\nu_2$, denoted $\nu_1\boxplus\nu_2$, such that $\nu_{A'_N+B_N'}\to\nu_1\boxplus\nu_2$ weakly, in probability, as $N\to\infty$, where $A'_N$ and $B'_N$ are random matrices chosen independently, with respective distributions $\rho_{A_N}$ and $\rho_{B_N}$. \endproclaim Thus, if we are given two $N\times N$ hermitian matrices (with $N$ large), and we only know their spectra, then we can bet, with a good chance to win, that the measure $\nu_{A+B}$ is well approximated by the measure $\nu_A\boxplus\nu_B$. 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 $\nu_A$ for arbitrarily large matrices, Theorem 1 introduces a binary operation $\boxplus$ 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 $\Cal O_A$, 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 $\xi_A$, then one has, denoting by $\mu*\nu$ the usual convolution of two probability measures on $\Bbb R$, \proclaim{Theorem 2} Let for each $N$, $A_N$ and $B_N$ be two real diagonal matrices, and let $\nu_1$ and $\nu_2$ be two probability measures with compact supports, such that $\nu_{A_N}\to\nu_1$ and $\nu_{B_N}\to\nu_2$ weakly as $N\to\infty$, then $\nu_{A'_N+B_N'}\to\nu_1*\nu_2$ weakly, in probability, as $N\to\infty$, where $A'_N$ and $B'_N$ are random matrices chosen independently, with respective distributions $\xi_{A_N}$ and $\xi_{B_N}$. \endproclaim 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 \cite{M}. Coming back to free convolution, note that we have not said how to compute explicitly the measure $\nu_1\boxplus\nu_2$ in terms of $\nu_1$ and $\nu_2$. 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. \head{3. Free random variables}\endhead Before we describe the connection with large matrices, we introduce the necessary notions from free probability. We start with some purely algebraic definitions. Let $\Cal A$ be a complex algebra with a unit, and $\varphi$ a $\Bbb C$-valued linear form on $\Cal A$, such that $\varphi(1)=1$. Although, in the examples that we shall consider, the algebra $\Cal A$ will always be non-commutative, it will be convenient to think of the elements of this algebra as random variables, while the map $\varphi$ 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. \proclaim{Definition 1} Let $I$ be a set of indices, and $\Cal B_i$, for $i\in I$, be subalgebras of $\Cal A$, containing the unit, then the algebras $\Cal B_i;i\in I$ are called free if one has $\varphi(a_1\ldots a_n)=0$ each time $\varphi(a_j)=0$ for all $j=1,\ldots ,n$ and $a_j\in \Cal B_{i_j}$ for some indices $i_1\not= i_2\not =\ldots \not=i_n$. \endproclaim Pursuing our analogy with classical probability, if we think of the algebras $\Cal B_i$ 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. \proclaim{Definition 2} Let $\Cal H_i;i\in I$ be subsets of $\Cal A$, they are called free if the subalgebras $\Cal B_i(=$ unital algebra generated by $\Cal H_i$), for $i\in I$, are free. \endproclaim In order to get acquainted with freeness we shall make a few computations. So let $\Cal B,\Cal C\subset \Cal A$ be two free subalgebras, and $b\in \Cal B$, $c\in \Cal C$, then we can write $b=b'+\bar b$ where $\bar b=\varphi(b)1$, so that $\varphi(b')=0$. Similarly we have $c=c'+\bar c$. Then using the definition of freeness, we see that $\varphi(b'c')=0$, thus $$\aligned\varphi(bc)&=\varphi((b'+\bar b)(c'+\bar c))\\ &=\varphi(b'c'+b'\bar c+\bar bc'+\bar b\bar c)\\ &=\varphi(b)\varphi(c)\endaligned$$ Here we have used $\varphi(\bar b c')= \varphi(\varphi(b)1.c')=\varphi(b)\varphi( c')=0$, and similarly $\varphi(b'\bar c)=0$. 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 $b_1,b_2\in \Cal B$, and $c_1, c_2\in \Cal C$, then we can expand $$\aligned\varphi(b_1c_1b_2c_2)&=\varphi((b_1'+\bar b_1)(c'_1+\bar c_1)( b_2'+\bar b_2)(c'_2+\bar c_2))\\ &=\varphi(b'_1c'_1b'_2c'_2+b'_1c'_1b'_2\bar c_2+b'_1c'_1\bar b_2c'_2+b'_1c'_1 \bar b_2\bar c_2+\\ &\qquad b'_1\bar c_1b'_2c'_2+b'_1\bar c_1b'_2\bar c_2 +b'_1\bar c_1\bar b_2c'_2+b'_1\bar c_1\bar b_2\bar c_2+\\ &\qquad\bar b_1 c'_1b'_2c'_2+\bar b_1 c'_1b'_2\bar c_2+\bar b_1 c'_1\bar b_2c'_2+ \bar b_1 c'_1\bar b_2\bar c_2+\\ &\qquad\bar b_1 \bar c_1b'_2c'_2+\bar b_1 \bar c_1b'_2\bar c_2 +\bar b_1 \bar c_1\bar b_2c'_2+\bar b_1 \bar c_1\bar b_2\bar c_2)\endaligned $$ Using the fact that $\varphi(b'_1c'_1b'_2c'_2)=0$ by the freeness property, we are left with terms in which some expectation factorizes. For example we shall treat the term $\varphi(b'_1\bar c_1 b'_2c'_2)=\varphi(c_1)\varphi(b'_1b'_2c'_2)$. By the factorization property already obtained one has $\varphi(b'_1b'_2c'_2)=\varphi(b'_1b'_2)\varphi(c'_2)=0$. The other terms can be treated by similar considerations, and after some straightforward manipulations we arrive at the formula $$\varphi(b_1c_1b_2c_2)=\varphi(b_1b_2)\varphi(c_1)\varphi(c_2)+ \varphi(b_1)\varphi(b_2)\varphi(c_1c_2)- \varphi(b_1)\varphi(b_2)\varphi(c_1)\varphi(c_2)$$ We observe that the computation of the expectation of the product $b_1c_1b_2c_2$ can be reduced to the computation of expectations in the subalgebras $\Cal B$ and $\Cal C$. 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 \proclaim{Proposition 1} Let $\Cal B_i;i\in I$ be free subalgebras in $\Cal A$, and $a_1,\ldots, a_n\in \Cal A$ such that for all $j=1,\ldots, n$, one has $a_j\in\Cal B_{i_j}$ for some $i_j\in I$. Let $\Pi$ be the partition of $\{1,\ldots, n\}$ determined by $j\sim k$ if $i_j=i_k$. For each partition $\pi$ of $\{1,\ldots, n\}$, let $\varphi_{\pi}=\prod_{\{j_1,\ldots ,j_r\} \in \pi\atop j_1<\ldots 0$, there exists a real number $\beta>0$ such that the function $G_{\mu}$ has a right inverse defined on the domain $\Theta_{\alpha,\beta}$, taking values in some domain of the form $$\Gamma_{\gamma,\lambda}=\{z=x+iy\,\vert\, y>0;-\gamma y< x<\gamma y;\vert z\vert\geq \lambda\}$$ with $\gamma,\lambda>0 $. Call $K_{\mu}$ this right inverse, and let $R_{\mu}(z)=K_{\mu}(z)-{1\over z}$. The function $R_{\mu}$ is called the $R$-transform of the measure $\mu$. Given another probability measure $\nu$ on $\Bbb R$, we shall define a new probability measure $\mu\boxplus\nu$ by the requirement that $$R_{\mu\boxplus \nu}=R_{\mu}+R_{\nu}$$ on some domain of the form $\Theta_{\alpha,\beta}$, 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 $\mu$ and $\nu$ have compact support. In fact, there is also an interpretation of $\mu\boxplus\nu$ as the distribution of the sum of two free (unbounded in general) self-adjoint elements affiliated to some non-commutative probability space, see \cite{BV} for details. \subhead 4.2 Law of large numbers\endsubhead The most general formulation of the law of large numbers for sums of independent equidistributed random variables asserts that the distribution of ${1\over n}(X_1+\ldots +X_n-M_n)$ converges weakly towards the Dirac measure at zero, for some constants $M_n$, if and only if the common distribution $\mu$ of the $X_j$ satisfies $t\mu(\Bbb R\setminus{[-t,t]})\to 0$ as $t\to\infty$. It turns out that exactly the same result holds if one replaces independent random variables by free ones. This is proved in \cite{BP}. \subhead 4.3 The Central Limit Theorem\endsubhead Let $\mu$ be a probability measure having zero mean and finite variance $\sigma$, then one has $R_{\mu}(z)\sim \sigma z$ as $z\to 0$. Let now $X_1,\ldots ,X_n, \ldots $ be a sequence of identically distributed free random variables, with common distribution $\mu$, then it is easy to see that the distribution of ${X_1+\ldots X_n\over \sqrt n}$ has $R$-transform $\sqrt n R_{\mu}({z\over \sqrt{n}})$, which converges, as $n\to\infty$, towards $\sigma z$. Using continuity properties of the $R$-tranform, it is not difficult to see that this implies the following free central limit theorem \proclaim{Theorem 6} The distribution of ${X_1+\ldots +X_n\over \sqrt n}$ converges weakly, as $n\to\infty$, towards the distribution with $R$-transform $\sigma z$.\endproclaim 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 ${1\over 2\pi \sigma}\sqrt{4\sigma-x^2}$ on the interval $[-2\sqrt \sigma, 2\sqrt \sigma]$. 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. \subhead 4.4 Infinitely divisible distributions\endsubhead There is a notion of infinitely divisible measures for the free additive convolution, which is the obvious one, namely a probability measure $\mu$ on the real line is said to be freely infinitely divisible if for every positive integer $n$, there exists a measure $\mu_n$ such that $\mu_n^{\boxplus n}=\mu$. The following characterization of freely infinitely divisible measures on $\Bbb R$ has been obtained by Bercovici and Voiculescu \cite{BV}. \proclaim{Theorem 7} A probability measure $\mu$, on $\Bbb R$ is infinitely divisible if and only if the function $R_{\mu}$ has an analytic continuation to the whole of $\Bbb C^+$, with values in $\Bbb C^-\cup\Bbb R$, and one has $$\lim_{y \rightarrow 0,y\in\Bbb R}yR_{\mu}( iy)=0$$ Furthermore any analytic function $R$ having the above properties is the $R$-transform of some probability measure.\endproclaim Functions such as those appearing in Theorem 7, have a Nevanlinna representation, which can be seen here as the free analogue of the L\'evy-Khintchine formula. More precisely, let $R$ be the $R$-transform of some infinitely divisible probability measure, then there exists a real number $\alpha$, and a finite positive measure $\nu$, on $\Bbb R$, such that $$R(z)=\alpha+\int_{-\infty}^{+\infty}{z+t\over 1-tz}d\nu(t)$$ The extreme points in this integral representation have an interpretation similar to the one of the classical L\'evy-Khintchine formula, namely the function $R(z)=\alpha$ is the $R$-transform of a Dirac mass at the point $\alpha$, the function $R(z)=\sigma z$, corresponding to a point mass at zero for $\nu$, is the $R$ transform of the semi-circular distribution with variance $\sigma$. Finally the probability measure with $R$-transform $R(z)=\lambda{z+t\over 1-tz}$ is the free analogue of the Poisson distribution, namely, it is the weak limit, as $n\to\infty$ of the measures $((1-{\lambda\over n})\delta_0+{\lambda\over n}\delta_t)^{\boxplus { n}}$ (the ``free binomial distributions''). \subhead 4.5 Stable distributions\endsubhead One can define stable distributions exactly as for classical convolution, namely, a probability measure on $\Bbb R$ is called stable, if and only if the set of probability measures obtained from $\mu$ by applying affine transformations of $\Bbb R$, is stable under free convolution. The set of all free stable probability measures on the real line has been determined by Bercovici and Voiculescu \cite{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 \roster \item $R(z)=e^{i\pi\theta}z^{\alpha-1}$ where $1<\alpha\leq 2$, and $(\alpha-2)\leq \theta\leq 0$. \item $R(z)=a+b\log z$ where $a\in\Bbb C^+\cup\Bbb R$ and $b\geq - \Im a/\pi$. \item $R(z)=e^{i\pi\theta}z^{\alpha-1}$ where $0<\alpha<1$, and $1\leq \theta\leq 1+\alpha$.\endroster In particular, the Cauchy distribution is a free stable distribution of stability index one. For these results, see \cite{BVB} \head 5. Speicher's combinatorial approach to freenes\endhead We shall now describe Speicher's combinatorial approach to the computation of the coefficients $c(\pi,\Pi)$ of Proposition 1, and some applications. A thorough discussion is given in \cite{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 $(i,j,k,l)\in S^4$, with $i