Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as [Formula: see text]-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an [Formula: see text] Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a ‘complex version’ of a [Formula: see text] submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

中文翻译：

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周期性随机矩阵系综的谱分布

Koloğlu、Kopp 和 Miller 计算了某类实数随机矩阵系综的极限谱分布，称为 [Formula: see text]-block 循环系综，并发现它完全等于 [Formula: see文本] 高斯酉系综。我们给出了一个更简单的证明，即在包含 Koloğlu–Kopp–Miller 研究的情况的非常一般的条件下，具有周期性对角线的实对称集合总是具有等于具有高斯项的有限 Hermitian 集合的特征值分布的极限谱分布，这是合奏的 [公式：见文本] 子矩阵的“复杂版本”。我们还证明了某些周期性有限 Hermitian 系综与高斯项之间的本质代数关系，之前的结果可以看作是实对称集合的渐近版本。证明表明，周期性随机矩阵系综和有限复 Hermitian 系综之间的这种一般对应在本质上是基本的和组合的。