We prove that the linear statistics of the eigenvalues of a Wigner matrix converge to a universal Gaussian process on all mesoscopic spectral scales, i.e. scales larger than the typical eigenvalue spacing and smaller than the global extent of the spectrum.

We prove a local law for the adjacency matrix of the Erdős-Rényi graph G (N, p) in the supercritical regime pN C log N where G(N, p) has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from [11] by extending them all the way down to the critical scale pN = O(log N ).A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed 2 and ∞ norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale pN = O(log N ). These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.

We present a simple and versatile method for deriving (an)isotropic local laws for general random matrices constructed from independent random variables. Our method is applicable to meanfield random matrices, where all independent variables have comparable variances. It is entirely insensitive to the expectation of the matrix. In this paper we focus on the probabilistic part of the proof -the derivation of the self-consistent equations. As a concrete application, we settle in complete generality the local law for Wigner matrices with arbitrary expectation.

We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph $${{\mathcal {G}}}(N,p)$$ G ( N , p ) . We show that if $$N^{\varepsilon } \leqslant Np \leqslant N^{1/3-\varepsilon }$$ N ε ⩽ N p ⩽ N 1 / 3 - ε 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 $$Np \geqslant N^{2/9 + \varepsilon }$$ N p ⩾ N 2 / 9 + ε down to the optimal scale $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε . The main technical achievement of our proof is a rigidity bound of accuracy $$N^{-1/2-\varepsilon } (Np)^{-1/2}$$ N - 1 / 2 - ε ( N p ) - 1 / 2 for the extreme eigenvalues, which avoids the $$(Np)^{-1}$$ ( N p ) - 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 $$Np \geqslant N^{\varepsilon }$$ N p ⩾ N ε .

Let H be a Hermitian random matrix whose entries Hxy are independent, centred random variables with variances Sxy = E|Hxy| 2 , where x, y ∈ (Z/LZ) d and d1. The variance Sxy is negligible if |x − y| is bigger than the band width W .For d = 1 we prove that if L W 1+ 2 7 then the eigenvectors of H are delocalized and that an averaged version of |Gxy(z)| 2 exhibits a diffusive behaviour, where G(z) = (H − z) −1 is the resolvent of H. This improves the previous assumption L W 1+ 1 4 of [9]. In higher dimensions d 2, we obtain similar results that improve the corresponding ones from [9]. Our results hold for general variance profiles Sxy and distributions of the entries Hxy.The proof is considerably simpler and shorter than that of [7,9]. It relies on a detailed Fourier space analysis combined with isotropic estimates for the fluctuating error terms. It is completely self-contained and avoids the intricate fluctuation averaging machinery from [7]. IntroductionGiven a large finite graph Γ, random band matrices H = (H xy ) x,y∈Γ are matrices whose entries H xy are independent and centred random variables and the variance S xy . .= E|H xy | 2 typically decays with the distance on a characteristic length scale W , called the band width of H.This name is due to the simplest one-dimensional model where Γ = {1, 2, . . . , N } and H xy = 0 if |x − y| W , where 1 W L. As an example of higher-dimensional models, one can take Γ to be the box of linear size L in Z d , so that the dimension of the matrix is N = L d . For a more general and extensive presentation of random band matrix models, we refer to [17].From the physics view point, random band matrices turn out to be very useful to study the disordered systems. In fact, it is conjectured that, depending on the level of energy and disorder strength, all these systems belong to two universality classes: in the strong disorder regime (as for the random Schrödinger operator models such as the Anderson model [1]), the eigenfunctions are localized and the local spectral statistics are Poisson, while in the weak disorder regime (as for the mean-field models such as Wigner matrices [18]), the eigenfunctions are delocalized and the local statistics are those of a mean-field Gaussian matrix ensemble.As W varies, random band matrices interpolate between these two classes: in particular, we recover the Wigner matrices by setting W = N and all variances equal, while for W = O(1) we essentially obtain the Anderson model. The delocalization property is expressed in the term of the localization length , which, in the framework of random matrices, describes the typical length scale of the eigenvectors of H: if the localization length is comparable with the system size, ∼ L, the system is delocalized and it is localized otherwise. The direct physical interpretation of the delocalization is that delocalized systems describe electric conductors, while localized systems insulators. Therefore, random band matrices represent a good model to investigate the Anderson metal-insulator phase transition....