We study the densities of limiting distributions of squared singular values of high-dimensional matrix products composed of independent complex Gaussian (complex Ginibre) and truncated unitary matrices which are taken from Haar distributed unitary matrices with appropriate dimensional growth. In the general case, we develop a new approach to obtain complex integral representations for densities of measures whose Stieltjes transforms satisfy algebraic equations of a certain type. In the special cases in which at most one factor of the product is a complex Gaussian, we derive elementary expressions for the limiting densities using suitable parameterizations for the spectral variable. Moreover, in all cases we study the behavior of the densities at the boundary of the spectrum.

中文翻译：

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高斯和截断酉随机矩阵乘积奇异值的谱密度

我们研究了由独立复高斯（复 Ginibre）和截断酉矩阵组成的高维矩阵乘积的平方奇异值的极限分布密度，截断酉矩阵取自具有适当维数增长的 Haar 分布酉矩阵。在一般情况下，我们开发了一种新方法来获得测量密度的复积分表示，其 Stieltjes 变换满足某种类型的代数方程。在乘积的至多一个因子是复高斯的特殊情况下，我们使用光谱变量的合适参数化来推导极限密度的基本表达式。此外，在所有情况下，我们都研究了光谱边界处的密度行为。