Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces

TL;DR

This paper develops the theory of second order freeness to describe the fluctuations of Gaussian and Wishart random matrices, introducing cyclic Fock space as an operator algebraic model for these fluctuations.

Contribution

It introduces the concept of second order freeness and provides a limit theorem for fluctuations of Gaussian and Wishart matrices using cyclic Fock space.

Findings

Gaussian and Wishart matrices are asymptotically free of second order.

Derived global fluctuation results for these matrices.

Established an operator algebraic model for fluctuations.

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 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.