Orthogonal Polynomials and Fluctuations of Random Matrices

Topological expansion for Haar-distributed orthogonal matrices and second-order freeness of orthogonally invariant ensembles

We present a genus expansion-type expression for the expected values of products of traces of expressions involving Haar-distributed orthogonal matrices. As with other real genus expansions, nonorientable surfaces appear, in addition to the orientable surfaces of the complex expansion. We use this expression to demonstrate that independent random matrices which are orthogonally in general position, such as matrices whose distributions are orthogonally invariant, are asymptotically real second-order free.

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].

正交不变随机矩阵的无穷小自由性

Infinitesimal freeness for orthogonally invariant random matrices

本文提出实无穷小自由性概念，证明正交不变随机矩阵在特定条件下渐近自由，并引入新的累积量定义及计算公式。

Almost Commuting Orthogonal Matrices

We show that almost commuting real orthogonal matrices are uniformly close to exactly commuting real orthogonal matrices. We prove the same for symplectic unitary matrices. This is in contrast to the general complex case, where not all pairs of almost commuting unitaries are close to commuting pairs. Our techniques also yield results about almost normal matrices over the reals and the quaternions.

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 asymptoticall…

Joint Global Fluctuations of complex Wigner and deterministic Matrices

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.