Spectral statistics of Erdős–Rényi graphs I: Local semicircle law

Abstract

We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, that is, 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. We prove that, as long as pN → ∞ (with a speed at least logarithmic in N ), the density of eigenvalues of the Erdős-Rényi ensemble is given by the Wigner semicircle law for spectral windows of length larger than N −1 (up to logarithmic corrections). As a consequence, all eigenvectors are proved to be completely delocalized in the sense that the ℓ ∞ -norms of the ℓ 2 -normalized eigenvectors are at most of order N −1/2 with a very high probability. The estimates in this paper will be used in the companion paper [Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues (2011) Preprint] to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pN ≫ N 2/3 .

Local Semicircle Law for Random Regular Graphs

“…, λ N are the eigenvalues of H. Theorem 1.1 implies that .17) with probability at least 1 − e −ξ log ξ . Following a standard application of the Helffer-Sjöstrand functional calculus along the lines of [20,Section 8.1], the following result may be deduced from (1.17). …”

“…The induction in the proof of Theorem 1.1 is not a continuity (or bootstrapping) argument, as used e.g. in the works [19][20][21] on local laws of models with independent entries. The multiplicative steps η → η/2 that we make are far too large for a continuity argument to work, and we correspondingly obtain much weaker a priori estimates from the induction hypothesis.…”

“…The multiplicative steps η → η/2 that we make are far too large for a continuity argument to work, and we correspondingly obtain much weaker a priori estimates from the induction hypothesis. Thus, our proof relies on a priori control of Γ instead of the error parameters Λ d and Λ o used in [19][20][21]. The advantage, on the other hand, of the approach taken here is that we only have to perform an order log N steps, as opposed to the N C steps required in bootstrapping arguments.…”

Delocalization for a class of random block band matrices

“…The proof is almost the same as the discussion on pages 2311-2312 in [10]. For the convenience of the reader, we sketch it below.…”

“…For the convenience of the reader, we sketch it below. At first, according to the discussion below (4.28) in [10], for any ı, j = 1, . .…”