1,670 research outputs found
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
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
Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
We extend the relation between random matrices and free probability theory
from the level of expectations to the level of fluctuations. We show how the
concept of "second order freeness", which was introduced in Part I, allows one
to understand global fluctuations of Haar distributed unitary random matrices.
In particular, independence between the unitary ensemble and another ensemble
goes in the large limit over into asymptotic second order freeness. Two
important consequences of our general theory are: (i) we obtain a natural
generalization of a theorem of Diaconis and Shahshahani to the case of several
independent unitary matrices; (ii) we can show that global fluctuations in
unitarily invariant multi-matrix models are not universal.Comment: 31 pages, new section on failure of universality added, typos
corrected, additional explanation
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