论文标题
相关矩阵的不均匀循环定律
Inhomogeneous Circular Law for Correlated Matrices
论文作者
论文摘要
我们考虑非hermitian随机矩阵$ x \ in \ mathbb {c}^{n \ times n} $，其条目之间的一般衰减相关性。对于大$ n $，经验光谱分布通过确定性密度很好地近似，该密度是根据对两个耦合非线性$ n \ times n $矩阵方程的系统的解决方案表示的。该密度被解释为在非交通概率空间上自由圆形元件与基质系数的线性组合的棕色度量。它是径向对称性的，在径向变量中的真实分析，并且在复杂平面的原点周围的磁盘上严格为阳性，而边缘处不连续下降至零。磁盘的半径是根据$ x $的条目的协方差明确给出的。我们以最佳的收敛速率显示了汇聚至局部光谱尺度的略高于典型的特征值间距。
We consider non-Hermitian random matrices $X \in \mathbb{C}^{n \times n}$ with general decaying correlations between their entries. For large $n$, the empirical spectral distribution is well approximated by a deterministic density, expressed in terms of the solution to a system of two coupled non-linear $n \times n$ matrix equations. This density is interpreted as the Brown measure of a linear combination of free circular elements with matrix coefficients on a non-commutative probability space. It is radially symmetric, real analytic in the radial variable and strictly positive on a disk around the origin in the complex plane with a discontinuous drop to zero at the edge. The radius of the disk is given explicitly in terms of the covariances of the entries of $X$. We show convergence down to local spectral scales just slightly above the typical eigenvalue spacing with an optimal rate of convergence.
