论文标题
圆形定律的波动，用于带有真实条目的随机矩阵
Fluctuation Around the Circular Law for Random Matrices with Real Entries
论文作者
论文摘要
我们将最近的结果[Cipolloni，Erdős，Schröder2019]扩展到非赫米特矩阵$ x $的线性特征值统计中的中心限制定理，具有独立的，相同分布的对称性类别。我们发现，期望和差异与它们的复杂对应物有很大不同，这反映了（i）真实矩阵在真实轴上的特殊光谱对称性； （ii）真实的I.I.D.矩阵具有许多真正的特征值。我们的结果概括了先前已知的特殊情况，即测试功能是分析性的[O'Rourke，Renfrew 2016]或矩阵元素的前四个矩与真正的高斯符合[Tao，Vu 2015; Kopel 2015]。证明的关键要素是分析几个弱依赖的戴森·布朗尼动议（DBMS）。与[Cipolloni，Erdős，Schröder2019]相比，实际情况的概念新颖性是，每个单独的DBM中随机差异的相关结构都是非平凡的，甚至可能危害其能力范围。
We extend our recent result [Cipolloni, Erdős, Schröder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [O'Rourke, Renfrew 2016] or the first four moments of the matrix elements match the real Gaussian [Tao, Vu 2015; Kopel 2015]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [Cipolloni, Erdős, Schröder 2019] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.
