Decompositions of the free additive convolution

Abstract

We introduce and study a new type of convolution of probability measures, denoted μ --ν and called the s-free additive convolution, which is defined by the subordination functions associated with the free additive convolution. We derive an alternating decomposition of μ --ν for compactly supported μ and ν, using another convolution called orthogonal additive convolution. This decomposition leads to two types of 'complete' alternating decompositions of the free additive convolution μ ν. More importantly, we develop an operatorial approach to the subordination property and introduce the associated notion of s-free independence. Moreover, we establish relations between convolutions associated with the main notions of noncommutative independence (free, monotone and boolean). Finally, our result leads to natural decompositions of the free product of rooted graphs.

Limit distributions of random matrices

“…However, we also show that by considering partial traces we can produce random matrix models for boolean independence, monotone independence and s-freeness. It is not a coincidence since all these notions of independence arise in the context of suitable decompositions of free random variables as shown in [21,22].…”

“…. , j k´1 P t1, 2u and k P N. The space F 1,2 is the s-free product of two free Fock spaces, pF 1 , ξ 1 q and pF 2 , ξ 2 q, the subspace of their free product pF 1 , ξ 1 q˚pF 2 , ξ 2 q (for the definition of the s-free product of Hilbert spaces and of the s-free convolution describing the free subordination property, see [21]). Now, observe that the action of δp1q onto M 2 can be identified with the action of the canonical noncommutative random variable γ 1 (built from ℓ 1 and its adjoint) restricted to F 1,2 .…”

Matricially free random variables

“…Finding simple analytic formulas for the corresponding four-parameter Cauchy transforms and densities does not seem possible in the general case. However, we shall derive decomposition formulas for those measures in terms of s-free additive convolutions [14], which gives some insight into their structure (see also [21] for recent results on the multivariate case). The s-free additive convolution refers to the subordination property for free additive convolution, discovered by Voiculescu [26] and generalized by Biane [4].…”

“…The s-free additive convolution refers to the subordination property for free additive convolution, discovered by Voiculescu [26] and generalized by Biane [4]. As shown in [14] and [15], there is a notion of independence, called freeness with subordination, or simply s-freeness, associated with the s-free additive convolution and its multiplicative counterpart.…”

“…Finally, recall after [1,14] that if (G 1 , e 1 ) and (G 2 , e 2 ) are two locally finite simple graphs and μ 1 and μ 2 are the associated spectral distributions, then the s-free product of G 1 and G 2 (in that order) can be interpreted as this half of the free product G 1 * G 2 which 'begins' (starting from the root) with a copy G 1 . Moreover, the associated spectral distribution is given by μ 1 μ 2 .…”