论文标题
中央限制定理的介质特征值统计数据的自由矩阵总和
Central limit theorem for mesoscopic eigenvalue statistics of the free sum of matrices
论文作者
论文摘要
我们考虑表格的随机矩阵$ h_n = a_n+u_n b_n u^*_ n $，其中$ a_n $，$ b_n $是$ n $ n $ n $确定性的Hermitian矩阵和$ u_n $的两个$ n $，是HAAR分布式的随机分布式unitaret unitation matrix。我们在常规频谱内所有介质尺度上为$ h_n $的线性特征值统计量建立了通用的中心限制定理。证明是基于研究线性特征值统计的特征功能，由两个主要步骤组成：（1）使用HAAR度量的左译本生成病房身份，以及用于$ h_n $的分辨率的本地法律以及$ h_n $的分析和分析质量卷积的特征，以衍生功能，以衍生功能，以衍生功能。 （2）使用HAAR度量的部分随机性分解来得出分解的两点产品函数的本地定律。我们还证明了正交结合的相应结果。
We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$, $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal Central Limit Theorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics, and consists of two main steps: (1) generating Ward identities using the left-translation-invariance of the Haar measure, along with a local law for the resolvent of $H_N$ and analytic subordination properties of the free additive convolution, allow us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.
