Abstract
We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph {{mathcal {G}}}(N,p). We show that if N^{varepsilon } leqslant Np leqslant N^{1/3-varepsilon } then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from Np geqslant N^{2/9 + varepsilon } down to the optimal scale Np geqslant N^{varepsilon }. The main technical achievement of our proof is a rigidity bound of accuracy N^{-1/2-varepsilon } (Np)^{-1/2} for the extreme eigenvalues, which avoids the (Np)^{-1}-expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for Np geqslant N^{varepsilon }.
Highlights
Introduction and main resultsLet A be the adjacency matrix of the Erdos–Rényi graph G(N, p)
To discuss the edge statistics of A in the sparse regime, we introduce the following conventions
If A is the normalized adjacency matrix (1.1) of G(N, p) from (1.3) and (1.6) we find that the condition 1 N p N 1/3 reads 1 q N 1/6, i.e. β ∈ (0, 1/6)
Summary
Let A be the adjacency matrix of the Erdos–Rényi graph G(N , p). Explicitly, A = (Ai j )iN, j=1 is a symmetric N × N matrix with independent upper triangular entries. Remark 1.5 is consistent with the fact that for more rigid graph models where the average degree is fixed, Z does not appear: for a random d-regular graph, the second largest eigenvalue of the adjacency matrix has Tracy–Widom fluctuations for N 2/9 d N 1/3 [2]. The central step of the proof is Proposition 4.1 below, which provides an upper bound for the fluctuations of the largest eigenvalue of H This is obtained by showing, for suitable E outside the bulk of the spectrum and η > 0, that the imaginary part of the Green’s function G(E + iη) ..= (H − E − iη)−1 satisfies Im Tr G(E + iη) 1/η.
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