Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

By James A. Mingo, Roland Speicher. 2004

146 citations recorded by the source corpus.

Abstract 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 interpret 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5c4e18f8fb8b5fcd3860b6f46451cf063b3bde57

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE (2007, unknown): “Fluctuations for Wigner and Wishart matrices have also been studied by Mingo and Speicher in [22] with the help of free probability tools.”
Fluctuations of eigenvalues and second order Poincaré inequalities (2007, background): “The recent advances in the theory of second order freeness (introduced by Mingo and Speicher [41]) are also of great interest.”
Second order asymptotics for matrix models (2006, background): “In the case m ≥ 2 but V = 0, the central limit theorem was obtained in [8, 13, 20].”
Free Probability and Random Matrices (2014, background): “The theory of second-order freeness, which was developed in [59, 128, 129], provides an analogous machinery for calculating the fluctuations of sums and products from those of the constituent matrices, provided one has asymptotic second order freeness.”
Statistical eigen-inference from large Wishart matrices (2007, background): “When Σ= I so that the SCM S has the (null) Wishart distribution, the eigenvectors of Q are the (appropriately normalized) Chebyshev polynomials of the second kind [Mingo and Speicher (2006)].”
Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices (2013, background): “Finally, but this is not really our concern here, the fluctuations of the trace of words in several random matrices were studied in [19,29,30,39].”
Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants (2006, background): “ave κ(1n,π)[a1,...,an] = 0 for all π ∈ S(n). Example 7.7. Let us see that this deﬁnition includes the deﬁnition of Voiculescu [VDN92] for (ﬁrst order) freeness and the deﬁnition of Mingo and Speicher [MS04] for second order freeness. 1) On the ﬁrst level this follows from the fact that our cumulants”
Polynomial convolutions and (nite) free probability (2021, background): “This included a theory of free cumulants, first introduced by Nica and Speicher, which gave a unified framework for understanding classical and free independence through the lens of combinatorics [26].”