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

Freeness and The Partial Transposes of Wishart Random Matrices

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.

多分体Wishart矩阵部分转置的中心极限定理

A central limit theorem for partial transposes of multipartite Wishart matrices

本文将量子信息中的部分转置方法推广到一般多分体Wishart矩阵，证明其几乎必然渐近自由性，并建立相应的中心极限定理。

Asymptotic Freeness of Random Permutation Matrices from Gaussian Matrices

We show that an independent family of uniformly distributed random permutation matrices is asymptotically *-free from an independent family of square complex Gaussian matrices and from an independent family of complex Wishart matrices, and that in both cases the convergence in *-distribution actually holds almost surely. An immediate consequence is that, if the rows of a GUE matrix are randomly permuted, then the resulting (non self-adjoint) random matrix has a *-distribution which is asymptotically circular; similarly, a random permutation of the rows of a complex Wishart matrix results in a …

Real second-order freeness and the asymptotic real second-order freeness of several real matrix ensembles

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, three real models of random matrices, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. We use a combinatorial approach to the matrix calculations similar to the genus expansion for complex random matrices, but in which nonorientable surfaces appear, demonstrating the commo…

On a class of free Levy laws related to a regression problem

The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Levy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.

实 Wishart 随机矩阵的渐近无穷小分布

The Asymptotic Infinitesimal Distribution of a Real Wishart Random Matrix

本文研究实 Wishart 矩阵的渐近无穷小矩，将其表示为两类平面图求和，并为涉及非交叉环形置换的第二项建立与高阶自由概率相关的递归公式。