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 \(N^{\varepsilon } \leqslant Np \leqslant N^{1/3-\varepsilon }\) 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 (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from \(Np \geqslant N^{2/9 + \varepsilon }\) down to the optimal scale \(Np \geqslant N^{\varepsilon }\). 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 Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for \(Np \geqslant N^{\varepsilon }\).

我们考虑一类稀疏随机矩阵，其中包括 Erdős–Rényi 图\({{\mathcal {G}}}(N,p)\)的邻接矩阵。我们证明如果\(N^{\varepsilon } \leqslant Np \leqslant N^{1/3-\varepsilon }\)那么所有远离 0 的非平凡特征值都具有渐近高斯涨落。这些波动由单个随机变量控制，该变量具有对图形总度数的解释。这将结果（Huang et al. in Ann Prob 48:916–962, 2020）从\(Np \geqslant N^{2/9 + \varepsilon }\)的极端特征值的波动扩展到最佳尺度\(Np \geqslant N^{\varepsilon }\)。我们证明的主要技术成就是精确度的刚性界限\(N^{-1/2-\varepsilon } (Np)^{-1/2}\)用于极端特征值，这避免了来自 Erdős et的\((Np)^{-1}\) -expansions阿尔。(Ann Prob 41:2279–2375, 2013), Huang 等人。(2020) 和 Lee 和 Schnelli（Prob Theor Rel Fields 171:543–616, 2018）。我们的结果是最后一个缺失的部分，添加到 Erdős 等人。(Commun Math Phys 314:587–640, 2012), He (稀疏随机矩阵的体特征值涨落.arXiv:1904.07140), Huang et al. (2020) 和 Lee 和 Schnelli (2018)，完整描述了\(Np \geqslant N^{\varepsilon }\)的稀疏随机矩阵的特征值波动。