# Fluctuations of extreme eigenvalues of sparse Erdős-Rényi graphs

Yukun He, Antti Knowles

Yukun He, Antti Knowles

**5/5/20**Published in : arXiv:2005.02254

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} |

.