On the global fluctuations of block Gaussian matrices

  • Mario DiazEmail author
  • James A. Mingo
  • Serban T. Belinschi


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.

Mathematics Subject Classification

60B20 46L54 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Centro de Investigación en Matemáticas A.C.GuanajuatoMexico
  2. 2.Queen’s UniversityKingstonCanada
  3. 3.CNRS - Institute of Mathematics of ToulouseToulouseFrance

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