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.

我们考虑非厄米随机矩阵 $X\in {\mathbb{C}}^{n\times n}$它们的条目之间具有普遍的衰减相关性。对于大n，经验谱分布很好地近似于确定性密度，用两个耦合非线性系统的解表示$n\times n$矩阵方程。该密度被解释为自由圆形元素与矩阵系数在非交换概率空间上的线性组合的布朗测度。它是径向对称的，在径向变量中是实解析的，并且在复平面中围绕原点的圆盘上严格为正，边缘处不连续下降为零。磁盘的半​​径是根据X的条目的协方差明确给出的。我们展示了收敛到略高于典型特征值间距的局部光谱尺度，并具有最佳收敛速度。