Mesoscopic eigenvalue statistics of Wigner matrices

He, Yukun; Knowles, Antti

doi:10.1214/16-aap1237

Cited by 78 publications

(114 citation statements)

References 20 publications

Supporting

Mentioning

112

Contrasting

Order By: Relevance

“…is the remainder term defined similarly to R K+1 in (3.2). Following a routine verification (one may refer to the proof of Lemma 4.6(i) in [13]), we see that for any D > 0, there is…”

Section: Extension To Non-gaussian Distribution and General Complex Casementioning

confidence: 83%

Diffusion Profile for Random Band Matrices: A Short Proof

Marcozzi

2019

J Stat Phys

Self Cite

View full text Add to dashboard Cite

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....

show abstract

“…is the remainder term defined similarly to R K+1 in (3.2). Following a routine verification (one may refer to the proof of Lemma 4.6(i) in [13]), we see that for any D > 0, there is…”

Section: Extension To Non-gaussian Distribution and General Complex Casementioning

confidence: 83%

Diffusion Profile for Random Band Matrices: A Short Proof

Marcozzi

2019

J Stat Phys

Self Cite

View full text Add to dashboard Cite

show abstract

“…The previous expression only depends on m 2 and so using the same argument as before we conclude the proof of (5.2). The proof for X ∈ C M ×(N −1) is omitted since is similar to the real case after replacing the cumulant expansion by its complex variant (Lemma 7.1 in [12]).…”

Section: Using the Resolvent Identity Gmentioning

confidence: 99%

Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices

Cipolloni

Erdős

2019

Random Matrices: Theory Appl.

View full text Add to dashboard Cite

We prove a central limit theorem for the difference of linear eigenvalue statistics of a sample covariance matrix [Formula: see text] and its minor [Formula: see text]. We find that the fluctuation of this difference is much smaller than those of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of [Formula: see text] and [Formula: see text]. Our result identifies the fluctuation of the spatial derivative of the approximate Gaussian field in the recent paper by Dumitru and Paquette. Unlike in a similar result for Wigner matrices, for sample covariance matrices, the fluctuation may entirely vanish.

show abstract

“…This allows us to have bounds for any high order cumulant term. In addition, we repeat the proof of Lemma 4.6(i) in [27] to estimate the remainder term. In this way we get a cumulant expansion with only the first few cumulant terms together with a "good" error term.…”

Section: Initial Expansion and The Leading Termmentioning

confidence: 99%

“….= −z − 2E G r+s for G r+s = (H − z) −1 . It is a routine verification (for details one can refer to Lemma 4.6(i) in [27]) to show that we can find L ≡ L(D + D , r, s) such that…”

Section: )mentioning

confidence: 99%

Mesoscopic eigenvalue density correlations of Wigner matrices

Knowles

2019

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

The celebrated Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory describes the universal correlations of eigenvalues on the microscopic scale, i.e. correlations of eigenvalue densities at points whose separation is comparable to the typical eigenvalue spacing. We investigate to what extent these statistics remain valid on mesoscopic scales, where the densities are measured at points whose separation is much larger than the typical eigenvalue spacing. In the mesoscopic regime, density-density correlations are much weaker than in the microscopic regime. More precisely, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. We obtain a precise and explicit formula for the two-point function. Among other things, this formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that the real symmetric terms contain subleading corrections matching precisely the WGMD statistics, while in the complex Hermitian case these subleading corrections are absent. We also uncover non-universal subleading correlations, which dominate over the universal ones beyond a certain intermediate mesoscopic scale. Our formula reproduces, in the extreme macroscopic regime, the well-known non-universal correlations of linear statistics on the macroscopic scale. Thus, our results bridges, within one continuous unifying picture, two previously unrelated classes of results in random matrix theory: correlations of linear statistics and microscopic eigenvalue universality. The main technical achievement of the proof is the development of an algorithm that allows to compute asymptotic expansions up to arbitrary accuracy of arbitrary correlation functions of mesoscopic linear statistics for Wigner matrices. Its proof is based on a hierarchy of Schwinger-Dyson equations for a sufficiently large class of polynomials in the entries of the Green function. The hierarchy is indexed by a tree, whose depth is controlled using stopping rules. A key ingredient in the derivation of the stopping rules is a new estimate on the density of states, which we prove to have bounded derivatives of all order on all mesoscopic scales.

show abstract

Mesoscopic eigenvalue statistics of Wigner matrices

Abstract: 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.

Cited by 78 publications

References 20 publications

Diffusion Profile for Random Band Matrices: A Short Proof

Diffusion Profile for Random Band Matrices: A Short Proof

Fluctuations for differences of linear eigenvalue statistics for sample covariance matrices

Mesoscopic eigenvalue density correlations of Wigner matrices

Contact Info

Product

Resources

About