Journal of Operator Theory

Volume 80, Issue 1, Summer 2018 pp. 47-76.

A combinatorial result on asymptotic independence relations for random matrices with non-commutative entries

Authors: Zhiwei Hao (1) and Mihai Popa (2)
Author institution: (1) School of Mathematics and Statistics, Central South University, Changsha 410083, China
(2) Department of Mathematics, University of Texas at San Antonio, One UTSA Circle San Antonio, Texas 78249, U.S.A. and ``Simion Stoilow'' Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Summary: The paper gives a class of permutations such that a semicircular matrix is free independent, or asymptotically free independent from the semicircular matrix obtained by permuting its entries. In particular, it is shown that semicircular matrices are asymptotically free from their transposes, a result similar to the case of Gaussian random matrices. There is also an analysis of asymptotic second order relations between semicircular matrices and their transposes, with results not very similar to the commutative (i.e.\ Gaussian random matrices) framework. The paper also presents an application of the main results to the study of Gaussian random matrices and furthermore it is shown that the same condition as in the case of semicircular matrices gives Boolean independence, or asymptotic Boolean independence when applied to Bernoulli matrices.

DOI: http://dx.doi.org/10.7900/jot.2017may21.2170
Keywords: free independence, semicircular variables, random matrices with non-commutative entries, Gaussian random matrices, Boolean independence

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