Free Probability of Type B: Analytic Interpretation and Applications

In this paper we give an analytic interpretation of free convolution of type B, and provide a new formula for its computation. This formula allows us to show that free additive convolution of type B is essentially a re-casting of conditionally free convolution. We put in evidence several aspects of this operation, the most significant being its apparition as an “intertwiner” between derivation and free convolution of type A. We also show connections between several limit theorems in type A and type B free probability. Moreover, we show that the analytical picture fits very well with the idea of considering type B random variables as infinitesimal deformations to ordinary non-commutative random variables. Free probability theory was introduced by D. Voiculescu in the eighties (see e.g. [23]). Already in his early work [22] Voiculescu has found analytical ways for computation of a number of natural operations in his theory, such as free convolution. Later on, Speicher [21] found that the combinatorics of free probability theory has to do with (“type A”) non-crossing partitions. For example, he found a description of the relation between moments and cumulants in terms of the lattice NC(n) of noncrossing partitions of {1, 2, . . . , n}; free independence could then be phrased in terms of vanishing of mixed cumulants. We refer the reader to [16] for a detailed description. This paper is devoted to the exploration of a new notion of independence, called type B freeness, which was introduced by Biane, Goodman and Nica in [6]. The original motivation for the introduction of this notion was the fact that the lattice of non-crossing partitions, central to (ordinary, or “type A”) probability theory is naturally associated to the symmetric groups, which are Weyl groups of Lie groups of type A. If the symmetric group is replaced by the hyperoctahedral group (the Weyl group of a type B Lie group), one obtains the lattice NC(n) of type B non-crossing partitions [19], and thus one seeks a probabilty theory whose underlying combinatorics is governed by this other lattice. In [6], the authors have shown that type B free probability theory does indeed exist: one can make sense of a type B law and of a type B non-commutative random variable. There is a notion of freeness, which has the desired combinatorial structure. Finally, one has the notions of type B free convolutions of type B laws, together with an appropriate linearizing transform (the R-transform of type B,) and so on. Later in [18], M. Popa showed that there is a natural notion of type B semicircular law and a type B central limit theorem. Type B freeness can be considered unusual in that there seemed to be no obvious notion of positivity. For a single random variable of type B, its law can be viewed as being described by a pair of measures (μ, μ′). Unfortunately, although it is clear that μ should be positive, there was no obvious positivity condition on μ′; and indeed, the measure μ′ associated to a type B semicircular variable need not be positive (as remarked in [18] for the central limit and Poisson limit distributions). To find a reasonable positivity assumption, we chose to introduce a notion of infinitesimal law of a family of random variables, which is a weakening of the notion of a type B law (this notion is almost implicit Date: March 15, 2009 ? Department of Mathematics and Statistics, University of Saskatchewan and Institute of Mathematics “Simion Stoilow” of the Romanian Academy. 106 Wiggins Road, Saskatoon, Saskatchewan S7N 5E6, CANADA. E-mail: [email protected]. Supported by a Discovery grant from NSERC Canada, and a University of Saskatchewan start-up grant. † Department of Mathematics, UCLA, Los Angeles, CA 90095, U.S.A. E-mail: [email protected]. Research supported by NSF grants DMS-0555680 and DMS-0900776. 1 2 S.T. BELINSCHI AND D. SHLYAKHTENKO in the work of Biane, Goodman and Nica; indeed, they show that type B probability is related to freeness with amalgamation over the algebra C+C~ of power series in ~ taken modulo terms of order ~ or higher). More precisely, there is an infinitesimal law associated to every family of type B random variables (though some information is lost when passing to the infinitesimal law). This weakening, however, is of no consequence in a single-variable case, and amounts to interpreting the pair (μ, μ′) as the zeroth and first derivative of a family of laws μt (i.e., μ = μ0 and μ′(f) = d dt ∣∣∣ t=0 μt(f) for sufficiently nice f). One natural notion of positivity is then to require existence of a family μt of positive laws whose derivative is μ′. One can then check that the obvious notion of infinitesimal freeness (which requires that freeness conditions are fulfilled to order o(t)) are compatible with type B freeness. In particular, it turns out (Theorem 28) that free convolution of type B is intimately related to free convolution of type A: if (η, η′) = (μ, μ′) B (ν, ν′) then η = μ ν and η′ = d dt (μt νt) (usual free convolution), where μt and νt are any two families of laws having as their derivatives at t = 0 μ′ and ν′, respectively. Although the notions of infinitesimal law, infinitesimal freeness add to the proliferation of different notions of non-commutative random variables, freeness and so on, we feel that these notions are justified since they simplify our presentation and look rather natural. For example, the rather mysterious type B semicircular law is nothing by the infinitesimal law associated to the family of laws of variables x+ ty where x and y are two free semcirculars (Example 33). The notion of infinitesimal freeness is in spirit related to the notion of second-order freeness introduced by Mingo and Speicher [14], but is different from it (our notion is related to a first derivative, and Speicher’s is related to a second derivative, defined in the case the first derivative vanishes). A very rich source of infinitesimal laws is given by random matrix theory. Indeed, if XN is an N × N random matrix, its moments typically have an expansion in powers of 1/N . Keeping the zeroth and first order terms in 1/N then gives rise to an infinitesimal law. Unfortunately, we were unable to find a direct connection between ordinary random matrices and type B freeness. The natural guess — taking XN to be a real Gaussian random matrix and looking at its law to order 1/N as N →∞ does not produce an infinitesimal (i.e., type B) semicirclar variable. Indeed, as was shown by [11, 9], the law of an N ×N real random matrix is approximately