arXiv 2203.087122022-03math.PR

Wigner矩阵特征值的更多极限分布

More Limiting Distributions for Eigenvalues of Wigner Matrices

Simona Diaconu

备注：: 23 pages

摘要：

特雷西-威多姆分布是概率论中最著名的定律之一，部分原因是它们与维格纳矩阵的联系。特别地，对于$A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ i.i.d.标准正态分布，其最大本征值$λ_1(A)$的波动由实值Tracy-Widom分布渐进描述$TW_1:$ $n^{2/3}(λ_1(A)-2) \Rightarrow TW_1.$正如经常发生的那样，高斯性可以放松，并且当$\mathbb{E}[a_{11}]=0, \mathbb{E}[a^2_{11}]=1,$和$a_{11}$的尾部衰减得足够快时，该结果成立：$\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=0,$，而当$a_{11}$的规律随指数$α\in (0,4),$有规律地变化时，对于缓慢变化的$c_a:(0,\infty) \to (0,\infty)$，$c_a(n)n^{1/2-2/α}λ_1(A)$收敛于Fréchet分布本文考虑一类边情形$\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=c \in (0,\infty),$，并揭示了$λ_1(A):$的一种新的极限行为，$λ_1(A):$是Fréchet分布的连续函数，其中$2,$（$λ_1(A)$在轻尾情形下的几乎必然极限）起着关键作用：$f(x)=\begin{cases} 2, & 0<x<1 \newline x+\frac{1}{x}, & x \geq 1 \end{cases}.$

英文摘要：

The Tracy-Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $A=\frac{1}{\sqrt{n}}(a_{ij})_{1 \leq i,j \leq n} \in \mathbb{R}^{n \times n}$ symmetric with $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue $λ_1(A)$ are asymptotically described by a real-valued Tracy-Widom distribution $TW_1:$ $n^{2/3}(λ_1(A)-2) \Rightarrow TW_1.$ As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[a_{11}]=0, \mathbb{E}[a^2_{11}]=1,$ and the tail of $a_{11}$ decays sufficiently fast: $\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=0,$ whereas when the law of $a_{11}$ is regularly varying with index $α\in (0,4),$ $c_a(n)n^{1/2-2/α}λ_1(A)$ converges to a Fréchet distribution for $c_a:(0,\infty) \to (0,\infty)$ slowly varying and depending solely on the law of $a_{11}.$ This paper considers a family of edge cases, $\lim_{x \to \infty}{x^4\mathbb{P}(|a_{11}|>x)}=c \in (0,\infty),$ and unveils a new type of limiting behavior for $λ_1(A):$ a continuous function of a Fréchet distribution in which $2,$ the almost sure limit of $λ_1(A)$ in the light-tailed case, plays a pivotal role: $f(x)=\begin{cases} 2, & 0<x<1 \newline x+\frac{1}{x}, & x \geq 1 \end{cases}.$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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