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

Abstract

Abstract. 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 exemple of relation of monotone independence arising from the study of Boolean independence.

Fluctuation moments induced by conjugation with asymptotically liberating random matrix ensembles

“…Evidence of the existence of second order behaviors, other than second order free independence and second order free independence in the real sense, is not new, at least, from an algebraic point of view. We mention in particular the papers [7] and [8] where the authors analyze fluctuation moments of matrices with entries from a possibly non-commutative unital algebra and obtain different relations from those mentioned above. Now, notice (1.1) alone is not enough to guarantee the existence of limiting second order behaviors in Theorem 4, in contrast to (1.10) and (1.11).…”

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

“…, even. The following result (see [7] and Theorem 22.3 from [14]) is a non-commutative version of the Wick Formula (see [6]) and will be utilized in the next sections: Theorem 2.3. (The free/Boolean Wick Formula) (i) If {s i } 1≤i≤m is a family of free semicircular non-commutative random variables of mean zero, and x j is a complex linear combination of s i for each j = 1, .…”

“…More precisely, it describes a class of permutations of the entries of a square matrix (the matrix transpose or the partial transpose from [1], [11] are just particular cases) with the following property: a semicircular, respectively Bernoulli matrix is (asymptotically) free, respectively (asymptotically) Boolean independent from the matrix obtained by permuting its entries. There is also a brief application of the results to the study of Gaussian random matrices and a detailed investigation of the second order relations between a semicircular matrix and its transpose, in the spirit of [9], [10] and [7], although the results are significantly different (see the more detailed description below). The methods employed are mostly combinatorial, heavily relying on the properties of permutations and non-crossing partitions.…”

“…The last part of the paper, Section 6, describes the second order independence relations between a semicircular matrix and its transpose, in the spirit of [9], [10] and [7]. We mention that the main results of this last section (see Theorem 6.7) are still different in nature from both the formulas in the commutative framework (see [10]) and the case of ensembles free semicircular random matrices (see [7]).…”

An asymptotic property of large matrices with identically distributed Boolean independent entries

“…Boolean probability theory have been in the literature at least since early 1970's (see [22]) with various developments, from stochastic differential equations to measure theory [18]. The topic has attracted an increasing interest in the recent years, such as the works Popa and Vinnikov [15] Gu and Skoufranis [2], Jiao and Popa [3], Liu [4,5] and Popa and Hao [14]. This is the motivation for the present paper, which studies the asymptotic behavior of random matrices with independent identically distributed entries in the framework of Boolean probability.…”