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 $\mathbb{G}(\mathit{N},\mathit{d}/\mathit{N})$ in the critical regime $\mathit{d}\asymp log\mathit{N}$ of the transition uncovered in (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 2141–2161; Ann. Probab. 47 (2019) 1653–1676), where the regimes $\mathit{d}\gg log\mathit{N}$ and $\mathit{d}\ll log\mathit{N}$ were studied. We establish a one-to-one correspondence between vertices of degree at least $2\mathit{d}$ and nontrivial (excluding the trivial top eigenvalue) eigenvalues of $\mathit{A}/\sqrt{\mathit{d}}$ 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 $\mathit{d}={\mathit{d}}_{\ast }=\frac{1}{log4-1}log\mathit{N}$. For $\mathit{d}<{\mathit{d}}_{\ast }$, we obtain rigidity bounds on the locations of all eigenvalues outside the interval $[-2,2]$, and for $\mathit{d}>{\mathit{d}}_{\ast }$, 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 (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 2141–2161). 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.