论文标题
随机矩阵产品的通用和最小值的值：一种简化的方法
Universality and least singular values of random matrix products: a simplified approach
论文作者
论文摘要
在本说明中，我们展示了如何在研究独立随机矩阵的局部普遍性时对特定翻译线性矩阵的最小奇异值的最小控制。该问题首先在Koppel，O'Rourke和VU的最新工作中考虑到，与他们的工作相比，我们的证明基本上更为简单，并且在更大的一般性中确定。特别是，我们仅假设合奏的条目是居中的，并且具有统一的$ 0 $和无穷大的第二和第四刻，而先前的工作则假定统一的subgaussian衰减条件，并且产品中每个因素内的条目都分布得很分布。 我们最不奇异的值结合的结果是，独立随机矩阵的产物的四动力匹配结果，最近由Koppel，O'Rourke和Vu获得，在较弱的假设下持有。我们的证明技术在研究结构化稀疏矩阵的研究中也具有独立的兴趣。
In this note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first considered in a recent work of Koppel, O'Rourke, and Vu, and compared to their work, our proof is substantially simpler and established in much greater generality . In particular, we only assume that the entries of the ensemble are centered, and have second and fourth moments uniformly bounded away from $0$ and infinity, whereas previous work assumed a uniform subgaussian decay condition and that the entries within each factor of the product are identically distributed. A consequence of our least singular value bound is that the four moment matching universality results for the products of independent random matrices, recently obtained by Koppel, O'Rourke, and Vu, hold under much weaker hypotheses. Our proof technique is also of independent interest in the study of structured sparse matrices.
