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The paper investigates the asymptotic behavior of (non-normalized) traces ofcertain classes of matrices with non-commutative random variables as entries.We show that, unlike in the commutative framework, the asymptotic behavior ofmatrices with free circular, respectively with Bernoulli distributed Booleanindependent entries is described in terms of free, respectively Booleancumulants. We also present an exemple of relation of monotone independencearising from the study of Boolean independence.

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