论文标题
稀疏Erdős-rényi图的极端特征值的波动
Fluctuations of extreme eigenvalues of sparse Erdős-Rényi graphs
论文作者
论文摘要
我们考虑一类稀疏的随机矩阵，其中包括Erdős-rényi图的邻接矩阵$ \ MATHCAL {G}（n，p）$。我们表明，如果$ n^{\ varepsilon} \ leq np \ leq n^{1/3- \ varepsilon} $，则所有非平凡的特征值远离0都具有渐近的高斯波动。这些波动受单个随机变量的控制，该变量的解释是图形的总数。这将结果[19]扩展到极端特征值的波动中，从$ np \ geq n^{2/9 + \ varepsilon} $下降到最佳尺度$ np $ np \ geq n^{\ varepsilon} $。我们证明的主要技术成就是准确性的刚度$ n^{ - 1/2- \ varepsilon} \，（np）^{ - 1/2} $用于极端特征值，避免了$（np）^{ - 1} $ - 从[9,19,24]扩展。我们的结果是添加到[8、12、19、24]中的最后一个缺失的部分，该作品对$ np \ geq n^{\ varepsilon} $的稀疏随机矩阵的特征值波动的完整描述。
We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős-Rényi graph $\mathcal{G}(N,p)$. We show that if $N^{\varepsilon} \leq Np \leq N^{1/3-\varepsilon}$ then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result [19] on the fluctuations of the extreme eigenvalues from $Np \geq N^{2/9 + \varepsilon}$ down to the optimal scale $Np \geq N^{\varepsilon}$. The main technical achievement of our proof is a rigidity bound of accuracy $N^{-1/2-\varepsilon} \, (Np)^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from [9,19,24]. Our result is the last missing piece, added to [8, 12, 19, 24], of a complete description of the eigenvalue fluctuations of sparse random matrices for $Np \geq N^{\varepsilon}$.
