论文标题
介镜中央限制非属性随机矩阵的定理
Mesoscopic Central Limit Theorem for non-Hermitian Random Matrices
论文作者
论文摘要
我们证明，eigenValues $ \ \ {σ_i\} _ i $ of time $ n \ times n $ n $ non-Hermitian随机矩阵，带有复杂的中心i.i.d.对于任何$ h^{2} _0 $ functions $ f $在任何点$ z_0 $ a BOST $ h^{2} _0 $ f $的任何介绍范围内的大部分频谱上的任何$ h^{2} _0 $ functions $ f $上都是渐近的高斯条目。这扩展了我们以前的结果[Arxiv：1912.04100]，该结果在宏观量表上是有效的，$ a = 0 $，以覆盖整个介质制度。主要新颖性是在频谱参数$ z_1，z_2 $的x $ hermitization产品的产物的本地法律，在整个中镜制度$ | z_1-z_2 | \ gg gg n^{ - 1/2} $中具有改进的错误术语。证明是动态的；它依赖于特征流方法的递归串联以及绿色函数比较想法与基础稳定性操作员不稳定模式的分离结合在一起。
We prove that the mesoscopic linear statistics $\sum_i f(n^a(σ_i-z_0))$ of the eigenvalues $\{σ_i\}_i$ of large $n\times n$ non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any $H^{2}_0$-functions $f$ around any point $z_0$ in the bulk of the spectrum on any mesoscopic scale $0<a<1/2$. This extends our previous result [arXiv:1912.04100], that was valid on the macroscopic scale, $a=0$, to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of $X$ at spectral parameters $z_1, z_2$ with an improved error term in the entire mesoscopic regime $|z_1-z_2|\gg n^{-1/2}$. The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator.
