We study the empirical spectral distribution (ESD) for complex n × n matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) n →∞ with k constant and (2) k →∞ with n bounded by O(kP) for some P > 0; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on k (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

中文翻译：

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复矩阵多项式的极限经验谱分布

我们研究了复杂的经验光谱分布 (ESD)n × n矩阵多项式ķ在对基础分布的相对温和的假设下，从而突出了普遍性现象。特别是，我们假设矩阵多项式的每个矩阵系数的条目具有均值零和有限方差，这可能允许不同系数的条目的不同分布。我们在两种不同的情况下得出了几乎可以肯定的 ESD 极限：（1）n →∞和ķ常数和 (2)ķ →∞和n被约束○(ķ磷)对于一些磷 > 0; 第二个结果还要求基础分布是连续的且均匀有界的。我们的结果是普遍的，因为它们取决于方差的选择，并且可能取决于ķ（如果它保持不变），但不是在基础分布上。通过固定方差，可以将结果专门用于特定模型，从而获得与特殊类别的标量多项式（例如 Kac、Weyl、椭圆和双曲多项式）已知的结果的矩阵多项式类似物。