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arXiv 1904.07985math.PRmath.CO
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稀疏Wigner矩阵谱中的野值

Outliers in spectrum of sparse Wigner matrices

Konstantin Tikhomirov, Pierre Youssef

备注：
Added reference to the related work arXiv:1905.03243
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摘要：

在本文中，我们研究了稀疏性对半圆律中离群点出现的影响。令$(W_n)_{n=1}^\infty$是随机对称矩阵的序列，使得每个$W_n$是在上面和在主对角线上具有与乘积$b_nξ$等分布的i.i.d项的$n\times n$，其中$ξ$是单位方差的实中心均匀有界随机变量，$b_n$是具有成功概率$p_n$的独立伯努利随机变量。假设$\lim\limits_{n\to\infty}n p_n=\infty$，我们表明，对于随机序列$(ρ_n)_{n=1}^\infty$由$$ρ_n:=θ_n+\frac{n p_n}{θ_n},\quad θ_n:=\sqrt{\max\big(\max\limits_{i\leq n}\|{\rm Row_i}(W_n)\|_2^2-np_n,n p_n\big)},$$的比率$\frac{\|W_n\|}{ρ_n}$收敛到一个概率。一个非中心对应的定理允许获得渐近表达式的Erdens-Renyi图，这是未知的政权$n p_n=Θ(\log n)$的特征值。特别地，由$A_n$表示$\mathcal{G}(n,p_n)$的邻接矩阵，由$λ_{|k|}(A_n)$表示其$k$第（绝对值）大的特征值，在假设$\lim\limits_{n\to\infty }n p_n=\infty$和$\lim\limits_{n\to\infty}p_n=0$的情况下，我们有：-（没有非平凡的离群值）如果$\liminf\frac{n p_n}{\log n}\geq\frac{1}{\log (4/e)}$，则对于任何固定的$k\geq2$，$\frac{|λ_{|k|}(A_n)|}{2\sqrt{n p_n}}$在概率上收敛到$1$。- （离群值）如果$\limsup\frac{n p_n}{\log n}<\frac{1}{\log (4/e)}$，则存在$\varepsilon>0$，使得对于任何$k\in\mathbb{N}$，我们都有$\lim\limits_{n\to\infty}\mathbb{P}\Big\{\frac{|λ_{|k|}(A_n)|}{2\sqrt{n p_n}}>1+\varepsilon\Big\}=1$。在概念层面上，我们的结果突出了稀疏矩阵谱中异常值的出现和变形Wigner矩阵中所谓的BBP相变现象的相似性。

英文摘要：

In this paper, we study the effect of sparsity on the appearance of outliers in the semi-circular law. Let $(W_n)_{n=1}^\infty$ be a sequence of random symmetric matrices such that each $W_n$ is $n\times n$ with i.i.d entries above and on the main diagonal equidistributed with the product $b_nξ$, where $ξ$ is a real centered uniformly bounded random variable of unit variance and $b_n$ is an independent Bernoulli random variable with a probability of success $p_n$. Assuming that $\lim\limits_{n\to\infty}n p_n=\infty$, we show that for the random sequence $(ρ_n)_{n=1}^\infty$ given by $$ρ_n:=θ_n+\frac{n p_n}{θ_n},\quad θ_n:=\sqrt{\max\big(\max\limits_{i\leq n}\|{\rm Row_i}(W_n)\|_2^2-np_n,n p_n\big)},$$ the ratio $\frac{\|W_n\|}{ρ_n}$ converges to one in probability. A non-centered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős--Renyi graphs, which were unknown in the regime $n p_n=Θ(\log n)$. In particular, denoting by $A_n$ the adjacency matrix of $\mathcal{G}(n,p_n)$ and by $λ_{|k|}(A_n)$ its $k$-th largest (by the absolute value) eigenvalue, under the assumptions $\lim\limits_{n\to\infty }n p_n=\infty$ and $\lim\limits_{n\to\infty}p_n=0$ we have: -(No non-trivial outliers) If $\liminf\frac{n p_n}{\log n}\geq\frac{1}{\log (4/e)}$ then for any fixed $k\geq2$, $\frac{|λ_{|k|}(A_n)|}{2\sqrt{n p_n}}$ converges to $1$ in probability. -(Outliers) If $\limsup\frac{n p_n}{\log n}<\frac{1}{\log (4/e)}$ then there is $\varepsilon>0$ such that for any $k\in\mathbb{N}$, we have $\lim\limits_{n\to\infty}\mathbb{P}\Big\{\frac{|λ_{|k|}(A_n)|}{2\sqrt{n p_n}}>1+\varepsilon\Big\}=1$. On a conceptual level, our result highlights similarities in appearance of outliers in spectrum of sparse matrices and the so-called BBP phase transition phenomenon in deformed Wigner matrices.

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