We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

中文翻译：

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非厄密随机矩阵线性特征值统计的中心极限定理

我们考虑具有复杂、独立、同分布中心项的大型非 Hermitian 随机矩阵X ，并证明其特征值的线性统计对于具有导数的测试函数是渐近高斯分布的。以前只有少数特殊情况才知道此结果；要么测试函数需要解析 [72]，要么矩阵元素的分布需要高斯分布 [73]，或者至少在前四个时刻匹配高斯分布 [82, 56]。我们发现限制方差对第四个累积量的确切依赖性是以前不知道的。该证明依赖于两个新颖的成分：(i) X的隐士化的两个解决方案的产品的本地法律具有不同的光谱参数和（ii）几个弱依赖的戴森布朗运动的耦合。这些方法也是我们在配套论文 [32] 中提出的关于实矩阵X的线性特征值统计的类似结果的关键输入。© 2021 作者。Communications on Pure and Applied Mathematics由 Wiley Periodicals LLC 出版。