论文标题
重尾wigner矩阵的有限等级扰动
Finite Rank Perturbations of Heavy-Tailed Wigner Matrices
论文作者
论文摘要
已经对Wigner矩阵的单级扰动进行了仔细的研究：让$ p = \ frac {1} {\ sqrt {\ sqrt {n}} a+θvv^t $ with $ a =（a_ {a_ {ij}}对称，$（a_ {ij}）_ {1 \ leq i \ leq j \ leq n} $ i.i.d.具有中心标准的正常分布，$θ> 0，v \ in \ mathbb {s}^{n-1}。$ $众所周知的$λ_1（p），$ $ p，$的最大特征值，$具有$θ_0= 1：$θ_0= 1：$θ\ feq leq 1，$ $ $ up uctar的$θ_0= 1：而对于$θ> 1，$ $λ_1（p）\ xrightArrow [] {A.S。}θ+θ^{ - 1}。$在更一般的条件下，$λ_1（p）的限制行为，$适当地归一化，也已确定：$ | v || v || v || v || v || v | | {如果$ v $集中在一个条目上，则$ a_ {11} $和高斯分布。这些融合需要有限的第四刻，本文认为情况违反了这种情况。 For symmetric distributions $a_{11},$ heavy-tailed with index $α\in (0,4),$ the fluctuations are shown to be universal and dependent on $θ$ but not on $v,$ whereas a subfamily of the edge case $α=4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a分别在[1，\ frac {128} {89}]中的$θ_0= 1，θ_0\的连续相变。这些结果建立在我们以前的基于$λ_1的渐近行为（\ frac {1} {\ sqrt {n}} a）$中的渐近行为。
One-rank perturbations of Wigner matrices have been closely studied: let $P=\frac{1}{\sqrt{n}}A+θvv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. with centered standard normal distributions, and $θ>0, v \in \mathbb{S}^{n-1}.$ It is well known $λ_1(P),$ the largest eigenvalue of $P,$ has a phase transition at $θ_0=1:$ when $θ\leq 1,$ $λ_1(P) \xrightarrow[]{a.s.} 2,$ whereas for $θ> 1,$ $λ_1(P) \xrightarrow[]{a.s.} θ+θ^{-1}.$ Under more general conditions, the limiting behavior of $λ_1(P),$ appropriately normalized, has also been established: it is normal if $||v||_{\infty}=o(1),$ or the convolution of the law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions $a_{11},$ heavy-tailed with index $α\in (0,4),$ the fluctuations are shown to be universal and dependent on $θ$ but not on $v,$ whereas a subfamily of the edge case $α=4$ displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is localized, each presenting a continuous phase transition at $θ_0=1, θ_0 \in [1,\frac{128}{89}],$ respectively. These results build on our previous which analyzes the asymptotic behavior of $λ_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.
