# Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices

@article{Benigni2017EigenvectorsDA,
title={Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices},
author={Lucas Benigni},
journal={arXiv: Probability},
year={2017}
}
• L. Benigni
• Published 19 November 2017
• Mathematics, Physics
• arXiv: Probability
We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. Our results apply to a generalized $N\times N$ Rosenzweig-Porter model. We prove that the eigenvectors entries are asymptotically Gaussian with a specific variance, localizing them onto a small, explicit, part of the spectrum. For a well spread initial spectrum, this variance profile universally follows a heavy-tailed Cauchy distribution. The proof relies on a…

## Figures from this paper

### Fluctuations in Local Quantum Unique Ergodicity for Generalized Wigner Matrices

• Mathematics
Communications in Mathematical Physics
• 2022
We study the eigenvector mass distribution for generalized Wigner matrices on a set of coordinates I, where N 6 |I| 6 N1−ε, and prove it converges to a Gaussian at every energy level, including the

### Normal fluctuation in quantum ergodicity for Wigner matrices

• Mathematics
The Annals of Probability
• 2022
We consider the quadratic form of a general deterministic matrix on the eigenvectors of an N×N Wignermatrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the largeN limit.

### Eigenstate Thermalization Hypothesis for Wigner Matrices

• Mathematics
Communications in Mathematical Physics
• 2021
We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to

### Extreme gaps between eigenvalues of Wigner matrices

• Mathematics
Journal of the European Mathematical Society
• 2021
This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the

### Central limit theorem for linear spectral statistics of deformed Wigner matrices

• Mathematics
• 2017
We consider large-dimensional Hermitian random matrices of the form $W=M+\vartheta V$ where $M$ is a Wigner matrix and $V$ is a random or deterministic, real, diagonal matrix whose entries are

### Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices

• Mathematics
• 2019
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal

### Optimal delocalization for generalized Wigner matrices

• Mathematics, Computer Science
• 2021

### Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrix

• Mathematics
• 2019
We consider $N$ by $N$ deformed Wigner random matrices of the form $X_N=H_N+A_N$, where $H_N$ is a real symmetric or complex Hermitian Wigner matrix and $A_N$ is a deterministic real bounded diagonal

### Eigenvector Statistics of L\'{e}vy Matrices

• Mathematics, Computer Science
• 2020
Statistics for eigenvector entries of heavy-tailed random symmetric matrices whose associated eigenvalues are sufficiently small show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue.

### Eigenvector statistics of Lévy matrices

• Mathematics, Computer Science
• 2021
Statistics for eigenvector entries of heavy-tailed random symmetric matrices whose associated eigenvalues are sufficiently small show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue.

## References

SHOWING 1-10 OF 65 REFERENCES

### The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

• Mathematics
• 2013
We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity

### Eigenvector distribution of Wigner matrices

• Mathematics
• 2011
We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure νij whose first two moments

### Bulk universality for Wigner matrices

• Mathematics
• 2009
We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk

### Universality of random matrices and local relaxation flow

• Mathematics
• 2009
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we

### Extreme gaps between eigenvalues of Wigner matrices

• Mathematics
Journal of the European Mathematical Society
• 2021
This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the

### 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

### Bulk universality for deformed Wigner matrices

• Mathematics
• 2016
We consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are

### Overlaps between eigenvectors of correlated random matrices

• Computer Science
Physical Review E
• 2018
We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in

### Fixed Energy Universality for Generalized Wigner Matrices

• Mathematics
• 2014
We prove the Wigner‐Dyson‐Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random

### Random matrices: Universal properties of eigenvectors

• Mathematics
• 2011
The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing)