Free Infinite Divisibility for $q$-Gaussians

Abstract

1. IntroductionIn this note we prove that the q-Gaussian distribution introduced byBozejk_ o and Speicher in [10] (see also the paper [9] of Bozejk_ o, Kummerer andSpeicher) is freely in nitely divisible when q2[0;1].We shall give a short outline for the context of this problem. In prob-ability theory, the class of in nitely divisible distributions plays a crucial role, inthe study of limit theorems, Levy processes etc. So it was a remarkable discoveryof Bercovici and Voiculescu [6] that there exists a corresponding class of freely in- nitely divisible distributions in free probability. These distributions are typicallyquite di erent from the classically in nitely divisible ones; for example, many ofthem are compactly supported. Nevertheless, work of numerous authors culminat-ing in the paper by [5] Bercovici and Pata showed that free ID distributions are ina precise bijection with the classical ones, this bijection moreover having numer-ous strong properties. For example, the semicircular law is the free analog of thenormal distribution. From this bijection, one might get the intuition that, perhapswith very rare exceptions such as the Cauchy and Dirac distributions, some mea-sures belong to the \classical world and some to the \free world. However, [4,Corollary 3.9] indicates that this intuition may be misleading: the normal distribu-tion, perhaps the most important among the classical ones, is also freely in nitelydivisible.One approach towards understanding the relationship between the clas-sical and free probability theories, and in particular the Bercovici-Pata bijection,have been attempts to construct an interpolation between these two theories. Theoldest such construction, due to Bozejk_ o and Speicher, is the construction of theq-Brownian motion. In particular, it provides a probabilistic interpretation for a(very classical) family of q-Gaussian distributions, which interpolate between thenormal (q= 1) and the semicircle (q= 0) laws. Probably the best known descrip-tion of the q-Gaussian distributions is in terms of their orthogonal polynomialsH