论文标题
Wigner矩阵特征值的更多限制分布
More Limiting Distributions for Eigenvalues of Wigner Matrices
论文作者
论文摘要
Tracy-Widom分布是概率理论中最著名的法律之一，部分原因是它们与Wigner矩阵的联系。特别是，对于$ a = \ frac {1} {\ sqrt {n}}}（a_ {ij}）_ {1 \ leq i，j \ leq n} \ in \ mathbb {r} n} $ i.i.d.标准正常，其最大特征值$λ_1（a）$的波动是由真实价值的tracy-widom分布渐近描述的$ \ mathbb {e} [a_ {11}] = 0，\ mathbb {e} [a^2_ {11}] = 1，$和$ a_ {11} $衰减的尾巴足够快：$ \ lim_ \ infty} {x^4 \ mathbb {p}（| a_ {11} |> x）} = 0，$，而当$ a_ {11} $定律定期随（0,4）中的索引$α\定期变化时$ c_a：（0，\ infty）\ to（0，\ infty）$慢慢变化，仅取决于$ a_ {11}的定律。 （0，\ infty），$并揭示了$λ_1（a）的一种新型限制行为：$ afréchet分布的连续函数，其中$ 2，$肯定的$λ_1（a）$在轻尾案件中扮演着重要的角色： $ f（x）= \ begin {cases} 2，＆0 <x <1 \ newline x+\ frac {1} {x}，＆x \ geq 1 \ end {case}。$
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: $f(x)=\begin{cases} 2, & 0<x<1 \newline x+\frac{1}{x}, & x \geq 1 \end{cases}.$
