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.

中文翻译：

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自由矩阵和的介观特征值统计的中心极限定理

我们考虑 $H_N=A_N+U_N B_N U^*_N$ 形式的随机矩阵，其中 $A_N$、$B_N$ 是两个 $N$ × $N$ 确定性 Hermitian 矩阵，$U_N$ 是 Haar 分布式随机酉矩阵矩阵。我们建立了一个通用的中心极限定理，用于在光谱的规则主体内的所有介观尺度上 $H_N$ 的线性特征值统计。该证明基于研究线性特征值统计的特征函数，包括两个主要步骤：（1）使用 Haar 测度的左平移不变性生成 Ward 恒等式，以及求解 $ H_N$ 和自由加性卷积的解析从属性质，使我们能够推导出特征函数的导数的显式公式；(2) 使用 Haar 测度的部分随机性分解推导出求解器的两点乘积函数的局部定律。我们还证明了正交共轭的相应结果。