Single eigenvalue fluctuations of sparse Erdős–Rényi graphs

RANDOM MATRIX

Speaker:

Yukun He - Zurich

I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{-1}\ll p \ll N^{-2/3}, 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. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1-\varepsilon}p^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Antti Knowles.

Zoom: https://harvard.zoom.us/j/98520388668?pwd=c1hVZk5oc3B6ZTVjUUlTN0J2dmdsQT09