# Bounds for the Stieltjes transform and the density of states of Wigner matrices

@article{Cacciapuoti2013BoundsFT, title={Bounds for the Stieltjes transform and the density of states of Wigner matrices}, author={Claudio Cacciapuoti and Anna V. Maltsev and Benjamin Schlein}, journal={Probability Theory and Related Fields}, year={2013}, volume={163}, pages={1-59} }

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from Erdős et al. (Adv Math 229(3):1435–1515, 2012; Electron J Probab 18(59):1–58, 2013), by removing the logarithmic corrections. As applications…

## 34 Citations

### Lectures on the local semicircle law for Wigner matrices

- Mathematics
- 2016

These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent…

### Local law and Tracy–Widom limit for sparse random matrices

- Mathematics, Computer Science
- 2016

A local law for the eigenvalue density up to the spectral edges is proved and it is proved that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations if a deterministic shift of the spectral edge due to the sparsity is included.

### Local Marchenko-Pastur law at the hard edge of the Sample Covariance ensemble

- Mathematics
- 2022

. Consider an N by N matrix X of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of X ∗ X converges to the Marchenko-Pastur law on the optimal scale…

### Bulk universality for generalized Wigner matrices with few moments

- Mathematics
- 2016

In this paper we consider $$N \times N$$N×N real generalized Wigner matrices whose entries are only assumed to have finite $$(2 + \varepsilon )$$(2+ε)th moment for some fixed, but arbitrarily small,…

### Local laws for non-Hermitian random matrices and their products

- MathematicsRandom Matrices: Theory and Applications
- 2019

We consider products of independent [Formula: see text] non-Hermitian random matrices [Formula: see text]. Assume that their entries, [Formula: see text], are independent identically distributed…

### Mesoscopic linear statistics of Wigner matrices

- Mathematics
- 2015

We study linear spectral statistics of $N \times N$ Wigner random matrices $\mathcal{H}$ on mesoscopic scales. Under mild assumptions on the matrix entries of $\mathcal{H}$, we prove that after…

### Local semicircle law under moment conditions. Part I: The Stieltjes transform

- Mathematics
- 2015

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ in which the upper triangular entries are independent identically distributed random variables with mean zero and unit variance.…

### On the local semicircular law for Wigner ensembles

- MathematicsBernoulli
- 2018

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being i.i.d. random variables with mean zero and unit variance. We additionally suppose that…

### Optimal Local Law and Central Limit Theorem for $$\beta $$-Ensembles

- MathematicsCommunications in Mathematical Physics
- 2022

In the setting of generic β-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the…

### Local semicircle law under moment conditions. Part II: Localization and delocalization

- Mathematics
- 2015

We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with upper triangular entries being independent identically distributed random variables with mean zero and unit variance. We…

## References

SHOWING 1-10 OF 37 REFERENCES

### Averaging Fluctuations in Resolvents of Random Band Matrices

- Mathematics, Computer Science
- 2013

A general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint is considered, which leads to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices and new delocalization bounds on the eigenvectors are established.

### Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

- Mathematics
- 2009

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under…

### Universality of Wigner random matrices: a survey of recent results

- Mathematics
- 2011

This is a study of the universality of spectral statistics for large random matrices. Considered are symmetric, Hermitian, or quaternion self-dual random matrices with independent identically…

### Universality for generalized Wigner matrices with Bernoulli distribution

- Mathematics
- 2010

The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the…

### Wegner estimate and level repulsion for Wigner random matrices

- Mathematics
- 2008

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive…

### Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge

- Mathematics
- 2010

This is a continuation of our earlier paper (Tao and Vu, http://arxiv.org/abs/0908.1982v4[math.PR], 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of…

### Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

- Mathematics
- 2009

We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix
is normalized so that the average spacing between consecutive eigenvalues is of order
$1/N$. We study the…

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

- Mathematics
- 2013

We consider the ensemble of adjacency matrices of Erdős–Renyi random graphs, that is, graphs on N vertices where every edge is chosen independently and with probability p≡p(N). We rescale the matrix…

### Random covariance matrices: Universality of local statistics of eigenvalues

- Mathematics
- 2012

We study the eigenvalues of the covariance matrix 1/n M∗M of a large rectangular matrix M = Mn,p = (ζij)1≤i≤p;1≤j≤n whose entries are i.i.d. random variables of mean zero, variance one, and having…