We complete the analysis of the extremal eigenvalues of the the adjacency matrix A of the Erdős-Rényi graph G(N,d/N) in the critical regime d \asymp \log N of the transition uncovered in [arXiv:1704.02953,arXiv:1704.02945], where the regimes d \gg \log N and d \ll \log N were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A / \sqrt{d} outside of the asymptotic bulk [−2,2]. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value d = d_* = \frac{1}{\log 4 - 1} \log N. For dd_* we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities.
Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara-Bass formula [arXiv:1704.02945]. Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of A and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the \ell^2-norms of the rows.
