## Final Report Summary - NCDFP (Non-Commutative Distributions in Free Probability)

Non-commutative distributions contain relevant information about several, in general non-commuting, operators. Typical examples of such operators are non-commuting random matrices. A main quantity of interest is to understand the eigenvalue distribution of various functions built out of those random matrices; in particular, when the size of the matrices gets large. Such questions, though a priori of theoretical nature, have quite some relevance for concrete applications, like in wireless networks or machine learning, where random matrices are used to model the transmission channels, and the eigenvalue distribution contains important information about the performance of the considered systems.

The non-commutativity of the involved operators moves this problem out of the realm of classical methods. Classical probability theory looks on similar questions, but since there the involved random variables commute, their distribution is a classical probability measure - a well-understood object, for which there exist many far-developed tools to study it. The non-commutative counterpart, which we study here, is much harder to grasp and we are in need of totally new methods to get an understanding of non-commutative distributions. The right mathematical frame for dealing with such questions is the setting of “free probability theory”.

In the ERC grant we made several break-throughs in such a theory of non-commutative distributions. One direction was to understand non-commutative distributions by their symmetries. Whereas in the classical setting symmetries are described by the concept of a “group”, in the non-commutative context one needs a more general version for this - the concept of a “quantum group”. Such quantum groups have been studied for quite a while by now in mathematics, but we were able to find, describe and classify new classes of such quantum groups, which are of great relevance for non-commutative distributions. Furthermore, we developed new analytic tools for deriving qualitative features of non-commutative distributions in a very general setting and we constructed and implemented new algorithms for the calculation of non-commutative distributions. Those computation algorithms go much beyond what was available before and will be instrumental in virtually any field where the asymptotic eigenvalue distribution of random matrices plays a role.

The non-commutativity of the involved operators moves this problem out of the realm of classical methods. Classical probability theory looks on similar questions, but since there the involved random variables commute, their distribution is a classical probability measure - a well-understood object, for which there exist many far-developed tools to study it. The non-commutative counterpart, which we study here, is much harder to grasp and we are in need of totally new methods to get an understanding of non-commutative distributions. The right mathematical frame for dealing with such questions is the setting of “free probability theory”.

In the ERC grant we made several break-throughs in such a theory of non-commutative distributions. One direction was to understand non-commutative distributions by their symmetries. Whereas in the classical setting symmetries are described by the concept of a “group”, in the non-commutative context one needs a more general version for this - the concept of a “quantum group”. Such quantum groups have been studied for quite a while by now in mathematics, but we were able to find, describe and classify new classes of such quantum groups, which are of great relevance for non-commutative distributions. Furthermore, we developed new analytic tools for deriving qualitative features of non-commutative distributions in a very general setting and we constructed and implemented new algorithms for the calculation of non-commutative distributions. Those computation algorithms go much beyond what was available before and will be instrumental in virtually any field where the asymptotic eigenvalue distribution of random matrices plays a role.