Freeness over the diagonal and global fluctuations of complex Wigner matrices

We characterize the limiting second order distributions of certain independent complex Wigner and deterministic matrices using Voiculescu’s notions of freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher’s notion of second order freeness gives a universal rule, in terms of marginal first and second order distribution. We adapt and reformulate this notion for operator-valued random variables in a second order probability space. The Wigner matrices are assumed to be permutation invariant with null pseudo variance and the deterministic matrices to satisfy a restrictive property