Abstract

Voiculescu’s notion of asymptotic free independence applies to a wide range of random matrices, including those that are independent and unitary invariant. In this work, we generalize this notion by considering random matrices with a tensor product structure that are invariant under the action of local-unitaries. We show that, given the existence of limit ‘tensor moments’ described by tuples of permutations, an independent family of local-unitary invariant random matrices satisfies a new kind of independence in the limit, which we will call ‘tensor freen independence’. This can be defined via vanishing mixed ‘tensor free cumulants’, allowing the joint moments of tensor free elements to be described in terms of that of individual elements. Additionally, we propose a tensor free version of the central limit theorem, which extends and recovers the recent results on central limit theorem for tensor products of free variables. This is joint work with Ion Nechita.