论文标题
关于全球和介绍线性特征值统计的广义矩阵的波动
On fluctuations of global and mesoscopic linear eigenvalue statistics of generalized Wigner matrices
论文作者
论文摘要
我们认为$ n $ by $ n $ real或复杂的通用wigner矩阵$ h_n $，其条目是独立的以均匀界限的为中心的随机变量。我们假设差异配置文件，$ s_ {ij}：= \ mathbb {e} | h_ {ij} |^2 $，满足$ \ sum_ {i = 1}^ns_ {ij {ij} = 1 $ 1 \ leq i，j \ leq n $，带有一些常数$ c \ geq 1 $。我们为全球尺度上的$ h_n $的线性特征值统计以及所有介质尺度上的线性特征值统计量建立了高斯波动，并在光谱边缘达到了所有介质尺度，其期望和方差根据方差概况符合。随后，我们分别获得了整体内部和边缘内线性特征值统计量的通用介质中心极限定理。
We consider an $N$ by $N$ real or complex generalized Wigner matrix $H_N$, whose entries are independent centered random variables with uniformly bounded moments. We assume that the variance profile, $s_{ij}:=\mathbb{E} |H_{ij}|^2$, satisfies $\sum_{i=1}^Ns_{ij}=1$, for all $1 \leq j \leq N$ and $c^{-1} \leq N s_{ij} \leq c$ for all $ 1 \leq i,j \leq N$ with some constant $c \geq 1$. We establish Gaussian fluctuations for the linear eigenvalue statistics of $H_N$ on global scales, as well as on all mesoscopic scales up to the spectral edges, with the expectation and variance formulated in terms of the variance profile. We subsequently obtain the universal mesoscopic central limit theorems for the linear eigenvalue statistics inside the bulk and at the edges respectively.
