Circular law for random block band matrices with genuinely sublinear bandwidth
Circular law for random block band matrices with genuinely sublinear bandwidth
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DOI：
10.1063/5.0042590
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发表时间：
2020-08
影响因子：
1.3
通讯作者：
Vishesh Jain;Indrajit Jana;K. Luh;Sean O’Rourke
中科院分区：
文献类型：
--
作者：
Vishesh Jain;Indrajit Jana;K. Luh;Sean O’Rourke
证明了一类具有真正次线性带宽的非厄米特随机分块带矩阵的循环定律。也就是说，证明了在(0，1)$中存在$\tau，使得如果矩阵$X$的带宽至少为$n^{1-\tau}$且非零项是均值为零且略大于4个有限矩的iid随机变量，则当适当归一化时，$X$的极限经验特征值分布在复平面中的单位圆盘上以概率收敛到均匀分布.关键的技术结果是具有真正次线性带宽的移位随机带块矩阵的一个最小奇异值上界，它改进了Cook在带矩阵设置{cook2018 Low}中的结果。
We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $\tau \in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-\tau}$ and the nonzero entries are iid random variables with mean zero and slightly more than four finite moments, then the limiting empirical eigenvalue distribution of $X$, when properly normalized, converges in probability to the uniform distribution on the unit disk in the complex plane. The key technical result is a least singular value bound for shifted random band block matrices with genuinely sublinear bandwidth, which improves on a result of Cook in the band matrix setting \cite{cook2018lower}.