Real second order freeness and Haar orthogonal matrices

Abstract

Abstract. We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier [r 1 , r 2 ].

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

“…The present paper comes as an addendum to these works, more in the spirit of [15]. More precisely, while [7], [8] and [9] study the behavior of important classes of random matrices with entries in a commutative algebra, we present some similar results for the case when the entries are not commuting.…”

“…Note that here the partitions are acting on sets of different lengths, due to the presence of terms of type tr(S(f is ) ks )I in the expressions of P s 's (and the analogues for Q s 's). But, according to (9), the factors of the type tr(S(f is ) ks ) and the partitions leaving invariant P s cancel each other, hence, with the notations from (7)…”

“…The fluctuations of traces of various classes of random matrices have been studied in the last two decades in both physics (see, for example [2], [4], [5]) and mathematics (see [7], [8] [9]) literature. Extensive works (such as [19], [18]) indicate that free independence is best suited to describe the interaction of important classes of independent ensembles of random matrices with respect to normalized traces.…”

“…It was shown that free independence and the corresponding Central Limit Theorem laws (centered semicircular distributions) behave in a very regular manner when tensoring with algebras of complex matrices ( [17]). In order to address interactions of independent ensembles of random matrices with respect to unnormalized traces (fluctuation moments, higher order trace-moments), recent works, such as [7] and [8], introduced the notion of second order freeness or the more refined real second order freeness ( [9]). The present paper comes as an addendum to these works, more in the spirit of [15].…”

Non‐crossing annular pairings and the infinitesimal distribution of thegoe

“…, A (N ) s , N × N matrices that have a joint limit tdistribution. Recall from [17] this means that {A distribution. Using our convention that A…”