论文标题
Wigner矩阵的功能性中心极限定理
Functional Central Limit Theorems for Wigner Matrices
论文作者
论文摘要
我们考虑了定期功能的波动$ f $的wigner矩阵$ w $被视为整个矩阵$ f（w）$。超越了所研究的奇特模式，$ \ mathrm {tr} [f（w）] $，它等同于特征值的习惯线性统计信息，我们表明$ \ m athrm {tr} [f（w）] $对于任何非框架界限的确定性确定性确定性的基质$ a $ is is asmptotty is asmptotty is asmptotty均正常。我们确定了这种波动的三种不同且渐近的独立模式，与奇特部分，无可蕾丝对角线部分和$ f（w）$的外部分子部分相对应，在整个介体状态下，我们发现，在那里，我们发现offiagonal模式比散文模式更小。此外，我们确定了本征态热假说中的波动[Deutsch 1991]，即证明特征功能与任何确定性基质的重叠是在较小的光谱平均后渐近高斯的。特别是，在宏观制度中，我们的结果概括[Lytova 2013]对复杂的$ W $以及两者之间的所有跨界合奏。主要的技术输入是最近的多回报的本地法律，该法律和伴侣论文[Cipolloni，Erdős，Schröder2020]中的无痕迹确定性矩阵。
We consider the fluctuations of regular functions $f$ of a Wigner matrix $W$ viewed as an entire matrix $f(W)$. Going beyond the well studied tracial mode, $\mathrm{Tr}[f(W)]$, which is equivalent to the customary linear statistics of eigenvalues, we show that $\mathrm{Tr}[f(W)]$ is asymptotically normal for any non-trivial bounded deterministic matrix $A$. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of $f(W)$ in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. In addition, we determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch 1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. In particular, in the macroscopic regime our result generalises [Lytova 2013] to complex $W$ and to all crossover ensembles in between. The main technical inputs are the recent multi-resolvent local laws with traceless deterministic matrices from the companion paper [Cipolloni, Erdős, Schröder 2020].
