Extremal eigenvalues of critical Erdős-Rényi graphs

Abstract

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 $d=d_{\ast }=\frac{1}{\log 4-1}\log N$. For $d<d_{\ast }$ we obtain rigidity bounds on the locations of all eigenvalues outside the interval $[-2,2]$, and for $d>d_{\ast }$ 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.

55 pages, 5 figures. We corrected some typos and small inconsistencies