论文标题
高斯一元复合矩阵多项式的极限经验光谱分布
The limit empirical spectral distribution of Gaussian monic complex matrix polynomials
论文作者
论文摘要
我们定义了具有可逆领导系数的随机矩阵多项式的经验光谱分布（ESD），并将其研究用于复杂的$ n \ times n $ n $ ghussian monic矩阵$ k $的多项式。在两种不同的情况下，我们获得了ESD几乎确定限制的确切公式：（1）$ n \ rightarrow \ infty $带有$ k $常数和（2）$ k \ rightarrow \ rightarrow \ infty \ infty $ at $ n $ constant。我们方法的主要工具是Tao，Vu和Krishnapur的替代原理。在此过程中，我们还发展了潜在独立利益的一些辅助结果：我们略微扩展了Bürgisser和Cucker的结果，该结果是针对非零平均矩阵的伪内膜标准的尾巴结合的，并且我们对某些结构性随机矩阵的奇异值获得了几个估计值。
We define the empirical spectral distribution (ESD) of a random matrix polynomial with invertible leading coefficient, and we study it for complex $n \times n$ Gaussian monic matrix polynomials of degree $k$. We obtain exact formulae for the almost sure limit of the ESD in two distinct scenarios: (1) $n \rightarrow \infty$ with $k$ constant and (2) $k \rightarrow \infty$ with $n$ constant. The main tool for our approach is the replacement principle by Tao, Vu and Krishnapur. Along the way, we also develop some auxiliary results of potential independent interest: we slightly extend a result by Bürgisser and Cucker on the tail bound for the norm of the pseudoinverse of a non-zero mean matrix, and we obtain several estimates on the singular values of certain structured random matrices.
