Optimal delocalization for generalized Wigner matrices

@article{Benigni2021OptimalDF,
  title={Optimal delocalization for generalized Wigner matrices},
  author={Lucas Benigni and Patrick Lopatto},
  journal={Advances in Mathematics},
  year={2021}
}
We study the eigenvectors of generalized Wigner matrices with subexponential entries and prove that they delocalize at the optimal rate with overwhelming probability. We also prove high probability delocalization bounds with sharp constants. Our proof uses an analysis of the eigenvector moment flow introduced by Bourgade and Yau (2017) to bound logarithmic moments of eigenvector entries for random matrices with small Gaussian components. We then extend this control to all generalized Wigner… 

Figures from this paper

Fluctuations in local quantum unique ergodicity for generalized Wigner matrices
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
Tail bounds for gaps between eigenvalues of sparse random matrices
We prove the first eigenvalue repulsion bound for sparse random matrices. As a consequence, we show that these matrices have simple spectrum, improving the range of sparsity and error probability
Thermalisation for Wigner matrices
We compute the deterministic approximation of products of Sobolev functions of largeWigner matricesW and provide an optimal error bound on their fluctuation with very high probability. This
Equipartition principle for Wigner matrices
Abstract We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a
Normal fluctuation in quantum ergodicity for Wigner matrices
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.

References

SHOWING 1-10 OF 61 REFERENCES
Sparse general Wigner-type matrices: Local law and eigenvector delocalization
We prove a local law and eigenvector delocalization for general Wigner-type matrices. Our methods allow us to get the best possible interval length and optimal eigenvector delocalization in the dense
Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices
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
Wegner estimate and level repulsion for Wigner random matrices
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
Eigenvector distribution of Wigner matrices
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
Local Semicircle Law and Complete Delocalization for Wigner Random Matrices
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
Delocalization of eigenvectors of random matrices with independent entries
Author(s): Rudelson, Mark; Vershynin, Roman | Abstract: We prove that an n by n random matrix G with independent entries is completely delocalized. Suppose the entries of G have zero means, variances
No-gaps delocalization for general random matrices
We prove that with high probability, every eigenvector of a random matrix is delocalized in the sense that any subset of its coordinates carries a non-negligible portion of its $${\ell_2}$$ℓ2 norm.
Bulk Universality for Wigner Matrices
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
Extreme gaps between eigenvalues of Wigner matrices
  • P. Bourgade
  • Mathematics, Physics
    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
High Dimensional Normality of Noisy Eigenvectors
We study joint eigenvector distributions for large symmetric matrices in the presence of weak noise. Our main result asserts that every submatrix in the orthogonal matrix of eigenvectors converges to
...
1
2
3
4
5
...