55 research outputs found
Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces
We extend the relation between random matrices and free probability theory
from the level of expectations to the level of fluctuations. We introduce the
concept of "second order freeness" and derive the global fluctuations of
Gaussian and Wishart random matrices by a general limit theorem for second
order freeness. By introducing cyclic Fock space, we also give an operator
algebraic model for the fluctuations of our random matrices in terms of the
usual creation, annihilation, and preservation operators. We show that
orthogonal families of Gaussian and Wishart random matrices are asymptotically
free of second order.Comment: 46 pages, 13 figures, second revision adds explanations, figures, and
reference
Real Second Order Freeness and Haar Orthogonal Matrices
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 [R1, R2].Comment: 50 pages, revision has refreshed references and corrected typo
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