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]. 

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References

SHOWING 1-10 OF 46 REFERENCES

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

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

The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

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

Fixed Energy Universality for Generalized Wigner Matrices

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

Extreme gaps between eigenvalues of Wigner matrices

  • P. Bourgade
  • 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

Thermalisation for Wigner matrices

Eigenstate Thermalization Hypothesis for Wigner Matrices

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

Quantum ergodicity on graphs: From spectral to spatial delocalization

We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrodinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrodinger

Functional Central Limit Theorems for Wigner Matrices

We consider the fluctuations of regular functions f of aWigner matrixW viewed as an entire matrix f(W ). Going beyond the well studied tracial mode, Tr f(W ), which is equivalent to the the customary

The behaviour of eigenstates of arithmetic hyperbolic manifolds

In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto