We characterize the limiting second order distributions of independent complex Wigner and deterministic matrices using Voiculescu’s notions of freeness over the diagonal. For unitary invariant random matrices, Mingo and Speicher’s notion of second-order freeness gives a universal rule to compute the global fluctuation. But this one is in general not valid for non-Gaussian Wigner matrices, since the fluctuations are not universal, depending in particular on the moment of order 4 of the matrices. Yet, it is possible to adapt Mingo-Speicher’s formulation and reformulate this notion for operator-valued random variables in a second-order probability space, and prove a universal rule for more general Wigner matrices (for which the marginal second-order distributions are not universal).
This video was produced by the University of Münster, and forms part of the workshop Non-commutative geometry meets topological recursion.
