Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

Abstract

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N ). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN ≫ N 2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments.

Convergence of Local Statistics of Dyson Brownian Motion

“…The Wigner-Dyson-Gaudin-Mehta conjecture, or the 'bulk universality' conjecture, states that the local eigenvalue statistics of Wigner matrices are universal in the sense that they depend only on the symmetry class of the random matrix ensemble (i.e., real symmetric or complex Hermitian) but are otherwise independent of the underlying law of the matrix entries. This conjecture has been established for all symmetry classes in the works [7,8,10,12,15,19]. Parallel results were obtained independently in various cases in [26,27].…”

“…In a paper with J. Huang [20] we prove that Erdős-Rényi graphs where the probability p of each edge occuring is as small as p ľ N ε {N exhibit bulk universality. The previous result obtained in [7,9] allowed for p only as small as p ľ N 2{3`ε {N .…”

Local Semicircle Law for Random Regular Graphs

“….= Φ −1 G ij . Then, by (3.27) and 18) assuming that the constant C p was chosen sufficiently large. This establishes the first estimate of (4.2).…”

Stability of the matrix Dyson equation and random matrices with correlations

“…The celebrated Wigner-Dyson-Mehta (WDM) universality conjecture, as formulated in the classical book of Mehta [44], asserts that the same gap statistics holds if the matrix elements are independent and have arbitrary identical distribution (they are called Wigner ensembles). The WDM conjecture has recently been proved in increasing generality in a series of papers [18,21,24,25] for both the real symmetric and complex hermitian symmetry classes via the Dyson Brownian motion. An alternative approach introducing the four-moment comparison theorem was presented in [49,50,52].…”