Freeness over the diagonal and global fluctuations of complex Wigner matrices

We characterize the possible limiting 2nd order distributions of certain independent complex Wigner and deterministic matrices thanks to Voiculescu's notions of operator-valued freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher's notion of 2nd order freeness gives a universal rule, in terms of marginal 1st and 2nd order distribution. We adapt and reformulate this notion for operator-valued random variables in a 2nd 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.