论文标题
带有真正sublinear带宽的随机块带矩阵的循环定律
Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth
论文作者
论文摘要
我们证明了具有真正sublinear带宽的一类非热块带矩阵的循环定律。也就是说，我们表明存在（0,1）$中的$τ\，因此，如果矩阵$ x $的带宽至少为$ n^{1-τ} $，而非零的条目为IID随机变量，则平均值零，略多于四个有限的时刻，则在限制的经验分布$ x $时，限制了$ x $的分布，并符合$ x $的分布，并符合$ x $的分布，并将在复杂平面中的磁盘。关键的技术结果是具有真正的sublinear带宽的移动随机带块矩阵的最小值值绑定，该矩阵在频带矩阵设置中的结果改善。
We prove the circular law for a class of non-Hermitian random block band matrices with genuinely sublinear bandwidth. Namely, we show there exists $τ\in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1-τ}$ 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.
