Transition from Tracy.Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erd.s.Rényi 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.s.Rényi graph G(N,p). Tracy.Widom fluctuations of the extreme eigenvalues for p.N.2/3 was proved in (Probab. Theory Related Fields 171 (2018) 543.616; Comm. Math. Phys. 314 (2012) 587.640). We prove that there is a crossover in the behavior of the extreme eigenvalues at p.N.2/3. In the case that N.7/9.p.N.2/3, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when p=CN.2/3, 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.s.Rényi graphs are less rigid than those of random d-regular graphs (Bauerschmidt et al. (2019)) of the same average degree.