Extremal eigenvalues and eigenvectors of deformed Wigner matrices
Extremal eigenvalues and eigenvectors of deformed Wigner matrices
复制标题
变形维格纳矩阵的极值特征值和特征向量
DOI：
10.1007/s00440-014-0610-8
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发表时间：
2013
影响因子：
2
通讯作者：
Kevin Schnelli
中科院分区：
文献类型：
--
作者：
J. Lee;Kevin Schnelli
考虑随机矩阵的形式$$H = W + \lambda V，\lambda \in {\mathbb {R}}^+$$H=W+λV，λ∈R+，其中$$W$$W是大小为$$N$$N的真实的对称或复Hermitian Wigner矩阵，$$V$$V是大小为$$N$$N的真实的有界对角随机矩阵，i.i.d.独立于$$W$$W的条目。我们假设$$W$$W的矩阵元素的分布是亚指数衰减的，我们选择$$\lambda \sim 1$$λ 1，使得$$W$$W和$$\lambda V$$λV的特征值通常具有相同的阶数。此外，我们假设$$V$$V的条目的密度在单个区间上被支持，并且在其支持的边缘附近是凸的。本文证明了在{\mathbb {R}}^+$$λ+∈R+中存在$$\lambda _+\使得$$H$$H的最大特征值在由$$V$$V的顺序统计量确定的大$$N$$N的极限内，其中$$\lambda >\lambda _+$λ>λ+.特别地，$$H$$H的最大特征值在极限$$N\rightarrow \infty $N→∞中具有威布尔分布，如果$$\lambda >\lambda _+$λ>λ+。此外，对于足够大的$$N$$N，我们证明了与最大特征值相关联的特征向量对于$$\lambda >\lambda _+$$λ>λ+是部分局部化的，而对于$$\lambda <\lambda _+$$λ是完全离域的
We consider random matrices of the form $$H = W + \lambda V, \lambda \in {\mathbb {R}}^+$$H=W+λV,λ∈R+, where $$W$$W is a real symmetric or complex Hermitian Wigner matrix of size $$N$$N and $$V$$V is a real bounded diagonal random matrix of size $$N$$N with i.i.d. entries that are independent of $$W$$W. We assume subexponential decay of the distribution of the matrix entries of $$W$$W and we choose $$\lambda \sim 1$$λ∼1, so that the eigenvalues of $$W$$W and $$\lambda V$$λV are typically of the same order. Further, we assume that the density of the entries of $$V$$V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $$\lambda _+\in {\mathbb {R}}^+$$λ+∈R+ such that the largest eigenvalues of $$H$$H are in the limit of large $$N$$N determined by the order statistics of $$V$$V for $$\lambda >\lambda _+$$λ>λ+. In particular, the largest eigenvalue of $$H$$H has a Weibull distribution in the limit $$N\rightarrow \infty $$N→∞ if $$\lambda >\lambda _+$$λ>λ+. Moreover, for $$N$$N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for $$\lambda >\lambda _+$$λ>λ+, while they are completely delocalized for $$\lambda <\lambda _+$$λ