TRANSITION FROM TRACY-WIDOM TO GAUSSIAN FLUCTUATIONS OF EXTREMAL EIGENVALUES OF SPARSE ERDOS-RENYI 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 Erdos-Renyi 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 similar to 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 Erdos-Renyi graphs are less rigid than those of random d-regular graphs (Bauerschmidt et al. (2019)) of the same average degree.