We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free convolution of the laws of $A$ and $B$ as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [3,4] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.

中文翻译：

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在规则边缘添加随机矩阵的光谱刚度

我们考虑两个大型 Hermitian 矩阵 $A$ 和 $B$ 的总和，通过 Haar 酉共轭将它们置于一般相对位置。我们证明，随着矩阵维数的增加，在略高于局部特征值间距的尺度上的特征值密度是由$A$和$B$定律的自由卷积渐近给出的。这意味着 Voiculescu 定理中特征值的最佳刚性和最佳收敛速度。我们之前的工作 [3,4] 在体光谱中建立了这些结果，当前的论文完全解决了光谱边缘的问题，前提是它们具有典型的平方根行为。我们证明的关键要素是通过正确考虑边缘附近局部密度的尖锐误差估计来补偿从属方程稳定性的恶化。