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On the global fluctuations of block Gaussian matrices

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Abstract

In this paper we study the global fluctuations of block Gaussian matrices within the framework of second-order free probability theory. In order to compute the second-order Cauchy transform of these matrices, we introduce a matricial second-order conditional expectation and compute the matricial second-order Cauchy transform of a certain type of non-commutative random variables. As a by-product, using the linearization technique, we obtain the second-order Cauchy transform of non-commutative rational functions evaluated on selfadjoint Gaussian matrices.

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Acknowledgements

We would like to thank R. Speicher for pointing out the possibility of using the matricial second-order Cauchy transform of block Gaussian matrices to obtain the second-order Cauchy transform of non-commutative rational functions evaluated on selfadjoint Gaussian matrices. We would also like to thank the anonymous reviewers and editors for their valuable comments and suggestions which improved the readability of the paper.

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Authors and Affiliations

  1. Centro de Investigación en Matemáticas A.C., Guanajuato, Mexico

    Mario Diaz

  2. Queen’s University, Kingston, Canada

    James A. Mingo

  3. CNRS - Institute of Mathematics of Toulouse, Toulouse, France

    Serban T. Belinschi

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  1. Mario Diaz
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  2. James A. Mingo
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  3. Serban T. Belinschi
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Correspondence to Mario Diaz.

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The work of Mario Diaz was partially supported by CIMI (Centre International de Mathématiques et d’Informatique) Excellence Program, ANR-11-LABX-0040-CIMI within the program ANR-11- IDEX-0002-02, while visiting the Institute of Mathematics of Toulouse, the Natural Sciences and Engineering Research Council of Canada, and an Ontario Trillium Scholarship. J. A. Mingo was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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Diaz, M., Mingo, J.A. & Belinschi, S.T. On the global fluctuations of block Gaussian matrices. Probab. Theory Relat. Fields 176, 599–648 (2020). https://doi.org/10.1007/s00440-019-00925-1

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