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Journal of Operator Theory

Volume 73, Issue 1, Winter 2015  pp. 71-90.

On fluctuations of traces of large matrices over a non-commutative algebra

Authors:  Yong Jiao (1) and Mihai Popa (2)
Author institution:(1) Institute of Probability and Statistics, Central South University, Changsha 410075, China
(2) University of Texas at San Antonio, Department of Mathematics, One UTSA Circle, San Antonio, TX 78249, U.S.A. and Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-014700, Romania

Summary: The paper investigates the asymptotic behavior of (non-normalized) traces of certain classes of matrices with non-commutative random variables as entries. We show that, unlike in the commutative framework, the asymptotic behavior of matrices with free circular, respectively with Bernoulli distributed Boolean independent entries is described in terms of free, respectively Boolean cumulants. We also present an example of relation of monotone independence arising from the study of Boolean independence.

Keywords: non-commutative random variables, random matrices, fluctuation moments, free independence, Boolean independence, monotone independence

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