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Journal of Operator Theory

Volume 85, Issue 1, Winter 2021  pp. 153-181.

Freeness over the diagonal and global fluctuations of complex Wigner matrices

Authors: Camille Male
Author institution: UMI Laboratorio Solomon Lefschetz & Centre National de la Recherche Scientifique, Cuernavaca, 62000, Mexico

Summary: We characterize the possible limiting $2^{\mathrm{nd}}$ 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 $2^{\mathrm{nd}}$ order freeness gives a universal rule, in terms of marginal $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ order distribution. We adapt and reformulate this notion for operator-valued random variables in a $2^{\mathrm{nd}}$ 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.

Keywords: free probability, large random matrices, freeness with amalgamation, central limit theorem

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