Operator-valued free multiplicative convolution: analytic subordination theory and applications to random matrix theory
We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an iteration process. This algorithm is easily numerically implementable. We present two concrete applications of our method: the product of two free operator-valued semicircular elements and the calculation of the distribution of for scalar-valued and , which are free. Comparision between the solution obtained by our methods and simulations of random matrices shows excellent agreement.
Work of J. Treilhard and C. Vargas was supported by funds from R. Speicher from the Alfried Krupp von Bohlen und Halbach Stiftung (”Rückkehr deutscher Wissenschaftler aus dem Ausland”) and from the DFG (SP 419/8-1), respectively.
Free probability is a quite recent theory that has gained interest in the last few years. One of its main applications is on the field of random matrices. More specifically, it provides a conceptual way of understanding the distribution of the eigenvalues of several large random matrices. The variety of matrix models where free probability can be used is growing in accordance to the developments of the theory.
The crucial requirement that allows the treatment of a random matrix model with the algebraic and analytical machinery of free probability is that the matrices involved should satisfy (asymptotically, as their size tends to infinity) freeness relations. Some of the most important random matrices, such as Wigner and Haar Unitary matrices, were shown, starting with the basic work , to have these freeness requirements among themselves and with respect to deterministic matrices.
The applicability of free probability increased rapidly, in different directions, with the implementation of Voiculescu’s operator-valued version of the theory. The main idea is that operator-valued freeness is a much less restrictive condition, but still most of the features of usual free probability theory are present. We are now able to work with block Gaussian matrices [18, 17], rectangular random matrices of different sizes  and more complicated combinations of random and deterministic matrices , where scalar-valued freeness breaks up.
Whereas the analytic theory of scalar-valued free convolutions is far evolved (and thus we have now a variety of ways to deal with asymptotic eigenvalue distributions of matrices which are free), the same cannot be said about the status of operator-valued convolutions. In particular, at the moment we do not have an analytic description of operator-valued convolutions which would be easily and controllably implementable on a computer. Thus, numerical investigations for the above mentioned random matrix models were done in a more or less ad hoc way. Our main aim is to improve on this situation and provide a coherent analytic description of operator-valued free convolution and show its usefulness for dealing with random matrix questions. In the present paper we concentrate on multiplicative free convolution, the question of additive free convolution will be addressed in another paper .
The present paper is motivated by the following problem (which was communicated to us by Aris Moustakas in this form in the context of wireless communications and, independently, by Raj Rao for the special case and in the context of random graphs): If and are free, then it is not true in general that the elements , are free. This has made the distribution of quite inaccessible up to now.
We observe, however, that the distribution of is the same (modulo a Dirac mass at zero) as the distribution of the element
which in turn has the same moments as
The advantage of this reformulation is that the matrices and are free with amalgamation over the algebra of constant matrices. Hence, the distribution that we are looking for will be given by first calculating the -valued free multiplicative convolution of and to obtain the -valued distribution of and then getting from this the (scalar-valued) distribution of by taking the trace over . This has motivated us to look at the general problem of dealing with operator-valued free multiplicative convolutions.
The standard way to deal with free multiplicative convolutions is through Voiculescu’s -transform. The generalization of the -transform to the operator-valued situation was found by Dykema . (A certain version of the -transform, via a Fock space-type construction, appeared already in the work of Voiculescu .) Direct computations of operator-valued -transforms for non-trivial elements are, however, extremely hard. Approximations can be done on domains corresponding to domains of the Cauchy-Stieltjes transform which are located far away from the real axis. For practical purposes this is a problem, since we are interested in the behavior of the Cauchy transform close to the real axis, in order to recover the distribution by an Stieltjes inversion.
Our main contribution to the theory is to find the subordination functions as iterative limits, similar to what has been done in the scalar case . We rely on the twisted multiplicative property of Dykema’s -transform.
This will allow us to set up fixed point equations to approximate effectively the values of the Cauchy transform of in the whole operator-valued upper half plane and in particular, close to the real axis. As inputs, we require the individual operator-valued Cauchy Transforms of and (or good approximations of these).
We want to stress that in the operator-valued context there are rarely situations where one has an analytic formula for the involved Cauchy transforms. The best one can hope for are equations which determine those transforms. In order to be applicable to concrete problems one needs of course a way to solve these equations - in particular, to single out the correct solution; our operator-valued equations usually have many solutions, only one of them corresponding to the wanted Cauchy transform. Since the characterization of the Cauchy transform among all solutions is by positivity requirements this is an intrinsic analytic characterization, which indicates that those equations cannot be solved by pure formal power series expansion arguments. So what we need for a theory of operator-valued convolution which is also practically applicable is to set up our theory in such a form which can also numerically be implemented (and such that the theory provides arguments for the working of this implementation). What has become more and more apparent in the scalar-valued context, namely that the subordination formulation of free convolution seems to be the right choice, is even more prominent in the operator-valued context. Trying to solve an operator-valued free multiplicative convolution problem directly with the help of the operator-valued -transform becomes quite a challenging task very soon, whereas using the subordination formulation, as presented in this paper, allows not only a satisfying analytic description of the involved transformations, but can also be implemented numerically very easily.
In Section 2, we will present our analytic description, via subordination, of the free multiplicative operator-valued convolution. In Section 3, we will show the usefulness of our approach by implementing our method to calculate the distribution of the product of two operator-valued free random variables and by comparing this with histograms for corresponding random models. We consider there two different types of examples: (i) the product of two operator-valued semicircular elements, the individual Cauchy transforms obtained by the fixed point equations described in ; (ii) in the context of our original problem described above, we treat the case , where is discrete and is either discrete or a semicircular variable.
2. Multiplication of operator-valued free random variables
We will call an operator-valued non-commutative probability space a triple , where is a von Neumann algebra, is a -subalgebra containing the unit of , and is a unit-preserving conditional expectation. Elements in will be called operator-valued (or -valued) random variables. The distribution of a random variable with respect to is, by definition, the set of multilinear maps
We call the moment of (or, equivalently, of ). It will be convenient to interprete as the constant equal to , the unit of (or, equivalently, of ) and , the expectation of . We denote by the -algebra generated by and the elements in .
Two algebras containing are called free with amalgamation over with respect to (or just free over ) if
whenever , satisfy and , . Two random variables are called free over if and are free over .
If are free over , then and depend only on and . Following Voiculescu, we shall denote these dependencies by and , and call them the free additive, respectively free multiplicative, convolution of the distributions and . It is known [20, 29] that both and are associative, but may fail to be commutative.
2.1. Analytic transforms
A very powerful tool for the study of operator-valued distributions is the generalized Cauchy-Stieltjes transform and its fully matricial extension [29, 30]: for a fixed , we define for all so that is invertible in . One can easily verify that is a holomorphic mapping on an open subset of . Its fully matricial extension is defined on the set of elements for which is invertible in , by the relation . It is a crucial observation of Voiculescu that the family encodes the distribution of . A succint description of how to identify the moment of when is known is given in .
In the following we will use the notation for the situation where and is invertible; note that this is equivalent to the fact that there exists a real such that . From the later it is clear that implies (because our conditional expectations are automatically completely positive).
From now on we shall restrict our attention to the case when are selfadjoint, and (for most applications) nonnegative. In this case, one of appropriate domains for - and the domain we will use most - is the operator upper half-plane Elements in this open set are all invertible, and is invariant under conjugation by invertible elements in . It has been noted in  that maps into the operatorial lower half-plane and has “good behaviour at infinity” in the sense that .
As, from an analytic function perspective, have essentially the same behaviour on for any , we shall restrict our analysis from now on to . However, all properties we deduce for this , and all the related functions we shall introduce, remain true, under the appropriate formulation, for all .
We shall use the following analytic mappings, all defined on ; all transforms have a natural Schwarz-type analytic extension to the lower half-plane given by ; in all formulas below, is fixed in :
the moment generating function:
The reciprocal Cauchy transform:
The eta transform (Boolean cumulant series):
In this paper we shall call this function “the h transform:”
Based on the moment generating function, Dykema  introduced the operator-valued version of Voiculescu’s -transform  as an analytic mapping on Banach algebras (an earlier, less easily employed version can be found in ). It is easy to note that , so that . Under the assumption that is invertible in (and, in particular, when ), the linear map becomes invertible, with inverse , and so by the usual Banach-space inverse function theorem, has an inverse around zero, which we shall denote by . The -transform is defined as
Dykema showed [13, Theorem 1.1] that, whenever and are both invertible in ,
2.2. Three ways to the subordination function
Here we shall describe three ways to finding the analytic subordination functions for free multiplicative convolution of operator-valued distributions. The analytic subordination has been proved in different contexts by Voiculescu and Biane [28, 9, 30, 31]. For the case , Biane showed that there exist analytic functions which preserve half-planes and satisfy . In particular, . Voiculescu extended this relation, essentially in , and made it more precise in , to the case of a general , under the assumption that is endowed with a tracial state and preserves this trace. Later, in , Biane’s subordination functions were found as limits of an iteration process involving and . A precise description of such an iterative process for (very general) positive operator-valued random variables, in the spirit of , will be our main contribution in this section.
Inspired by the shape of formula (8), we claim that for our purposes, the most appropriate writing of the operator-valued subordination phenomenon is the following:
Let be two random variables with invertible expectations, free over . There exists a Gâteaux holomorphic map such that
and are analytic around zero;
For any so that , the map , is well-defined, analytic and
for any fixed .
Moreover, if one defines , then
Following the proof of this theorem, we shall mention several conditions under which some hypotheses, particularly the - rather inconvenient in practical applications - invertibility requirement, can be weakened or entirely dropped.
We shall split our proof in several remarks, formulas and lemmas. For our purposes, a slight variation of the -transform will be more useful: we define the sigma transform , again on a neighbourhood of zero. Elementary arithmetic manipulations show that and so . Using this relation we write on a neighbourhood of zero in :
Now define , again for sufficiently small. We substitute in the previous relation for to obtain
Recalling from equation (6) the definition of the h transform, we obtain
An application of on both sides gives , or
This shows us that exists on a small enough neighbourhood of the origin, and is a fixed point for the map introduced in Theorem 2.2. Moreover, this indicates that is analytic around zero (a fact that follows quite easily also from the definition of as ). The most significant, however, is the fact that, under the conditions of analyticity of the two maps and around zero, equation (9) uniquely determines , and thus a function satisfying (9) must also satisfy , and vice-versa.
Now we observe that, under the additional assumption of the existence of a trace on so that , this function coincides with a function provided by Voiculescu: in , Voiculescu proved that, whenever and are free over , the range of the analytic map is included in . ( denotes the conditional expectation with onto the von Neumann algebra generated by and .) It follows quite easily that the range of the map is also included in . We claim that
Since up to this moment has only been defined on a neighbourhood of zero, we only need to verify the equality for small. Indeed, the above relation is equivalent to , by the bimodule property of . Inverting both sides of the equality and applying the conditional expectation gives We invert, take a as a factor and subtract 1 in order to get . As it was noted by Voiculescu , and we shall argue below, is invertible whenever . Since the set is open for any , and , it follows, by analytic continuation, that there exists an so that
Until now we have argued that an analytic map satisfying parts (1) and (2) of our theorem exists and is unique (i) on a neighbourhood of zero for any -valued non-commutative probability space , and (ii) that this extends analytically to when is endowed with a tracial state so that remains a trace. We go now to the third way of identifying , namely as a fixed point of an analytic mapping, method which will allow us to extend to a set of the form for any type of non-commutative probability space .
Assume that is invertible in and . If is so that , then .
For simplicity, let us replace by , so that
We shall split our problem in two and use the same method as in : assume that is an arbitrary positive linear functional on so that . We define
As , and , are all strictly positive, it follows that , and, since is positive and faithful, . Thus, , and, in particular, invertible. Since , the Cauchy-Schwarz inequality tells us that whenever . We take
The Nevanlinna representation [1, Chapter III] allows us to write the inequality for all . Equality holds at one point of if and only if .Taking in the above we get
Since this inequality holds for all states on , we conclude that
We shall now prove that :
The last inequality is trivially true by functional calculus and the invertibility of . Putting the inequalities together gives
Since this is true for all positive , we get that maps into itself.
Next, we make use again of the same trick: we define
As before, whenever (the case , for example, is not excluded here), for any . We take again limit as of to obtain . The same argument used above implies that , and so, for we obtain , for all . This implies , as claimed. ∎
By the definition of the h transform (6), we have . The previous lemma allows us to write
i.e. for any , and with . Thus, we have shown that lands strictly into whenever and
It has been shown in  that for any and . By noting that if and only if , we obtain that and for any .
We can now improve on our previous statement: for , and so that ,
i.e. for any , and with .
The functions h associated to selfadjoints have convergent power series expansions around the origin. Indeed,
which gives the power series espansion of around zero and shows that .
We note that, as shown in the above Remark, there exists a so that are defined on and
Choosing with guarantees that and , so maps into . Thus, by the Earle-Hamilton Theorem [12, Theorem 11.1], has a unique attracting fixed point in which we shall denote by , and
exists for all , . The correspondence is clearly analytic, being a uniform limit of analytic maps. Moreover, on a small enough neighbourhood of zero, by the uniqueness of the fixed point guaranteed by the Earle-Hamilton Theorem.
For our fixed given in the statement of our theorem, let us fix an as above. Consider the set
This set is clearly open and connected (we can find the element in both open sets whose union we considered). The above indicates that
given by is analytic;
given by is analytic.
The union of the two domains of mentioned above is again connected, as we immediately note by identifying the point in both sets. Since the following chain of implications holds,
the strict inclusion
holds whenever .
Next, for any positive linear functional on and so that , we shall show that is bounded. The argument uses the fact that the set is convex. we choose , , and consider . The map lands, for a fixed , entirely in , independently of . Thus, there exists a small enough simply connected complex neighbourhood of , which does not depend on so that still lands in for all . We obtain that all maps in the family
take values in . This means that the family is normal (as a family of functions between complex domains). On the other hand, for very small, we know that
which, in particular, means that exists and is finite. This, together with the above argued normality, implies that exists as a holomorphic function from the given neighbourhood of to . Now any linear functional on has a unique Jordan decomposition as a linear combination of four positive linear functionals. Thus, for any in the dual of , the family is bounded. The uniform boundedness principle (see, for example, [23, Lemma 1]) guarantees that is bounded. The fact that is a von Neumann algebra implies that must have a w-convergent subsequence. Since , it is clear that any such limit point must belong to