Extremal eigenvalues of critical Erdős–Rényi graphs

Abstract

We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdős-Rényi graph G(N, d/N) in the critical regime d log N of the transition uncovered in [2, 3], where the regimes d log N and d 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/ √ 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∗ = 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 [3]. 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 `2-norms of the rows.