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May 2022 Normal fluctuation in quantum ergodicity for Wigner matrices
Giorgio Cipolloni, László Erdős, Dominik Schröder
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Ann. Probab. 50(3): 984-1012 (May 2022). DOI: 10.1214/21-AOP1552

Abstract

We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×N Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large N limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by Marcinek and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).

Acknowledgments

L.E. would like to thank Zhigang Bao for many illuminating discussions in an early stage of this research. The authors are also grateful to Paul Bourgade for his comments on the manuscript and the anonymous referee for several useful suggestions.

Citation

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Giorgio Cipolloni. László Erdős. Dominik Schröder. "Normal fluctuation in quantum ergodicity for Wigner matrices." Ann. Probab. 50 (3) 984 - 1012, May 2022. https://doi.org/10.1214/21-AOP1552

Information

Received: 1 March 2021; Revised: 1 September 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

Digital Object Identifier: 10.1214/21-AOP1552

Subjects:
Primary: 60B20
Secondary: 15B52

Keywords: Dyson Brownian motion , eigenvector moment flow , Local law , stochastic eigenstate equation

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.50 • No. 3 • May 2022
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