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

Abstract

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically B-diagonal and Boolean independent from each other. The paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that random matrices considered are asymptotically Boolean independent from their partial transposes. The main results of the paper are based on combinatorial techniques.Mathematics Subject Classification: Primary 60B20; Secondary 46L53, 06A07.

The free tangent law

“…This kind of product can be traced back to the work of Bożejko [13]. For more details about the Boolean probability theory see the standard references [14,49,37,3,27,41,42,47].…”

“…Speicher and Woroudi [49] introduced another concept of probability, namely Boolean probability. In recent years, a number of papers [49,37,3,27,41,42,47] have investigated theorems for the Boolean convolution of probability measures. The key concept of this definition is the notion of noncommutative Boolean independence.…”

Answer to a question by A. Mandarino, T. Linowski and K. Życzkowski

“…To prove (24), we shall use Lemma 4.4(iv) the multiplicative property of the leading term of the Wiengarten function, as mentioned in Preliminaries.…”

“…The techniques employed here to answer the question above track back to [22], where general entry permutations (including the map µ N from above) are studied in the framework of Gaussian random matrices. Ideas from [22] were further developed in [24], [23] for matrices with non-commutative entries, in the new and interesting work [3] for Wiegner matrices, in [18] for Haar unitaries. [18] gives general necessary conditions, for a sequence of permutations σ N N such that U σ N N is asymptotically circular and free from U N N ; furthermore, Consequence ... from [18] shows that U µ N N 2 N is asymptotically circular distributed and free from U N 2 N , thus giving a partial answer to the question from [14].…”

Special matrices used in the analysis of the Boolean quadratic forms

“…Speicher and Woroudi [10] introduced a new concept of probability. Then many, new theorems appeared about the Boolean probability ( [1], [2], [4], [6], [7], [8], [9]). In [3] Ejsmont and Hęćka show that the limit of mixed sums of commutators and anti-commutators…”