In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that each W_n is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product , where 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 , we show that for the random sequence given by , the ratio converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime . In particular, denoting by A_n the adjacency matrix of the Erdős–Renyi graph and by its kth largest (by the absolute value) eigenvalue, under the assumptions and we have (1) (No non‐trivial outliers): if then for any fixed k ≥ 2, converges to 1 in probability; and (2) (Outliers): if then there is ε > 0 such that for any , we have . On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices.

中文翻译：

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稀疏Wigner矩阵谱的离群值

在本文中，我们研究了稀疏性对半圆定律中离群值出现的影响。令其为随机对称矩阵的序列，使得每个W _n为n  ×  n，且iid项位于主对角线的上方和上方，且均乘积为，其中是单位方差的实心均匀有界随机变量，b _n是独立的伯努利成功概率为p _n的随机变量。假设，我们证明对于由 给出的随机序列，比率收敛到一个概率。定理的非中心对应项允许获得Erdős-Renyi图的特征值的渐近表达式，这在该系统中是未知的。特别是，表示由甲_Ñ鄂尔多斯-仁义图的邻接矩阵，并通过其ķ次最大（由绝对值）本征值，假设下和有（1）（没有非平凡离群值）：如果 那么对于任何固定ķ  ≥2 ，收敛于1概率; 和（2）（离群值）：如果则存在ε  > 0，使得对于任何，我们有。从概念上讲，我们的结果揭示了稀疏矩阵谱中异常值的出现与变形的Wigner矩阵中的所谓BBP相变现象的相似性。