# Normal fluctuation in quantum ergodicity for Wigner matrices

@article{Cipolloni2022NormalFI, title={Normal fluctuation in quantum ergodicity for Wigner matrices}, author={Giorgio Cipolloni and L'aszl'o ErdHos and Dominik Schroder}, journal={The Annals of Probability}, year={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. The proof is a combination of the energy method for the Dyson Brownian motion inspired by [24] and our recent multi-resolvent local laws [11].

## 8 Citations

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

- MathematicsCommunications 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…

### Rank-uniform local law for Wigner matrices

- Mathematics, Computer Science
- 2022

It is proved that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix has approximately Gaussian fluctuation for the bulk spectrum.

### Equipartition principle for Wigner matrices

- Physics, MathematicsForum of Mathematics, Sigma
- 2021

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…

### Dynamical Localization for Random Band Matrices up to $W\ll N^{1/4}$

- Mathematics
- 2022

We consider a large class of N ×N Gaussian random band matrices with band-width W , and prove that for W ≪ N they exhibit Anderson localization at all energies. To prove this result, we rely on the…

### Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

- Mathematics
- 2022

We consider Hermitian random band matrices H = (hxy) on the d-dimensional lattice (Z/LZ)d, where the entries hxy = hyx are independent centered complex Gaussian random variables with variances sxy =…

### Extremal statistics of quadratic forms of GOE/GUE eigenvectors

- Mathematics, Computer Science
- 2022

It is proved that, as long as the deterministic matrix has rank much smaller than √ N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians.

### Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble

- Physics
- 2022

Entanglement entropy is a powerful tool in characterizing universal features in quantum many-body systems. In quantum chaotic Hermitian systems, typical eigenstates have near maximal entanglement…

### A Local Wheeler-DeWitt Measure for the String Landscape

- Physics
- 2022

According to the ‘Cosmological Central Dogma’, de Sitter space can be viewed as a quantum mechanical system with a ﬁnite number of degrees of freedom, set by the horizon area. We use this assumption…

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