Transition from Tracy-Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erd\H{o}s-R\'enyi graphs

We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R'enyi graph $G(N,p)$. Tracy-Widom fluctuations of the extreme eigenvalues for $p\gg N^{-2\mathrm{/}3}$ was proved in [17,46]. We prove that there is a crossover in the behavior of the extreme eigenvalues at $p\sim N^{-2/3}$\mathrm{.}Inthecasethat$N^{-7/9}\ll p\ll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=C{N}^{-2\mathrm{/}3}$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy-Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse Erd\H{o}s-R'enyi graphs are less rigid than those of random $d$-regular graphs [4] of the same average degree.