Statistical properties of structured random matrices

On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.

结构化随机矩阵与循环累积量：一种自由概率方法

Structured random matrices and cyclic cumulants: A free probability approach

本文提出一类由四个基本性质刻画的大型结构化随机矩阵，证明其在矩阵值及逐点非线性运算下稳定，并基于极值化问题给出计算子块谱的高效方法，通过组合与代数证明验证其与自由概率的联系。

对“结构化随机矩阵与循环累积量：一种自由概率方法”的补充

Addition to "Structured random matrices and cyclic cumulants: A free probability approach"

本文精化随机矩阵系综定义，扩展第四公理处理不相交循环累积量，并证明非线性变换下的稳定性定理仍成立。

Statistical properties of eigenvectors and eigenvalues of structured random matrices

We study the eigenvalues and the eigenvectors of $N\times N$ structured random matrices of the form $H = W\tilde{H}W+D$ with diagonal matrices $D$ and $W$ and $\tilde{H}$ from the Gaussian Unitary Ensemble. Using the supersymmetry technique we derive general asymptotic expressions for the density of states and the moments of the eigenvectors. We find that the eigenvectors remain ergodic under very general assumptions, but a degree of their ergodicity depends strongly on a particular choice of $W$ and $D$. For a special case of $D=0$ and random $W$, we show that the eigenvectors can become crit…

Random unitary matrices associated to a graph

We analyze composed quantum systems consisting of $k$ subsystems, each described by states in the $n$-dimensional Hilbert space. Interaction between subsystems can be represented by a graph, with vertices corresponding to individual subsystems and edges denoting a generic interaction, modeled by random unitary matrices of order $n^2$. The global evolution operator is represented by a unitary matrix of size $N=n^k$. We investigate statistical properties of such matrices and show that they display spectral properties characteristic to Haar random unitary matrices provided the corresponding graph…

结构化随机矩阵的奇异性度

Singularity degree of structured random matrices

本文研究结构化厄米随机矩阵的态密度奇异性，通过分类零子矩阵相对位置，确定密度爆发指数即奇异性度。