Free probability of type B: analytic interpretation and applications

TL;DR

This paper provides an analytic interpretation and new computational formula for free convolution of type B, revealing its connection to conditionally free convolution and its role as an intertwiner between derivation and free convolution of type A.

Contribution

It introduces an analytic framework for free convolution of type B, linking it to conditionally free convolution and exploring its applications in limit theorems and infinitesimal deformations.

Findings

Derived a new formula for free convolution of type B.

Showed the connection between type B and conditionally free convolution.

Illustrated the role of type B as an intertwiner between derivation and free convolution.

Abstract

In this paper we give an analytic interpretation of free convolution of type B, introduced by Biane, Goodman and Nica, 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.