Sparse random matrices: Spectral edge and statistics of rooted trees
Sparse random matrices: Spectral edge and statistics of rooted trees
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DOI：
10.1017/s0001867800010661
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发表时间：
2001-03-01
影响因子：
1.2
通讯作者：
Khorunzhy, A
中科院分区：
文献类型：
--
作者：
Khorunzhy, A
继Furedi和Komlos之后，我们发展了一种图论方法来研究具有独立元素的大型随机矩阵的高阶矩。我们将这种方法应用于稀疏N x N随机矩阵A（N. p），平均每行有p个非零元素。我们的一个结果与平行于A（N，p）的谱范数在极限1中的渐近性态有关，其中A（N，p）的谱范数在极限1中的渐近性态远小于p远小于N。我们证明了p（c）= logN是lim parallel to A（N，p）/rootp parallel to有界与否的临界值.讨论了这一结果与Erdos-Renyi极限定理及大随机图的性质之间的关系。在证明中，主要问题是，即使当k ->无穷大时，k条边的平面根树的平均顶点度仍然有界。这一观察意味着相当精确的估计的时刻A（N.p）他们导致某些推广的结果西奈和Soshnikov的普遍性的地方谱统计的边界上的极限谱的大型随机矩阵。
Following Furedi and Komlos, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N x N random matrices A(N. p) that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm parallel toA(N,p)parallel to in the limit 1 much less than p much less than N. We show that the value p(c) = log N is the critical one for lim parallel to A(N,p) /rootp parallel to to be bounded or not. We discuss relations of this result with the Erdos-Renyi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k --> infinity. This observation implies fairly precise estimates for the moments of A(N.p) They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.