Extremal eigenvalues of critical ErdH{o}s-R\'enyi graphs

We complete the analysis of the extremal eigenvalues of the the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi 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 $d < d_*$ we obtain rigidity bounds on the locations of all eigenvalues outside the interval $[-2,2]$, and for $d > d_*$ 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.