article published yes Giorgio Cipolloni author 42198EFA-F248-11E8-B48F-1D18A9856A870000-0002-4901-7992 László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Dominik J Schröder author 408ED176-F248-11E8-B48F-1D18A9856A870000-0002-2904-1856 LaEr department 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). Institute of Mathematical Statistics2022 eng 0091-1798 2168-894X 2103.06730 00079396340000510.1214/21-AOP1552 503984-1012 Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2022). Normal fluctuation in quantum ergodicity for Wigner matrices. <i>Annals of Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/21-AOP1552">https://doi.org/10.1214/21-AOP1552</a> Cipolloni, Giorgio, et al. “Normal Fluctuation in Quantum Ergodicity for Wigner Matrices.” <i>Annals of Probability</i>, vol. 50, no. 3, Institute of Mathematical Statistics, 2022, pp. 984–1012, doi:<a href="https://doi.org/10.1214/21-AOP1552">10.1214/21-AOP1552</a>. G. Cipolloni, L. Erdös, and D. J. Schröder, “Normal fluctuation in quantum ergodicity for Wigner matrices,” <i>Annals of Probability</i>, vol. 50, no. 3. Institute of Mathematical Statistics, pp. 984–1012, 2022. Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Normal Fluctuation in Quantum Ergodicity for Wigner Matrices.” <i>Annals of Probability</i>. Institute of Mathematical Statistics, 2022. <a href="https://doi.org/10.1214/21-AOP1552">https://doi.org/10.1214/21-AOP1552</a>. Cipolloni G, Erdös L, Schröder DJ. 2022. Normal fluctuation in quantum ergodicity for Wigner matrices. Annals of Probability. 50(3), 984–1012. Cipolloni G, Erdös L, Schröder DJ. Normal fluctuation in quantum ergodicity for Wigner matrices. <i>Annals of Probability</i>. 2022;50(3):984-1012. doi:<a href="https://doi.org/10.1214/21-AOP1552">10.1214/21-AOP1552</a> G. Cipolloni, L. Erdös, D.J. Schröder, Annals of Probability 50 (2022) 984–1012. 114182022-05-29T22:01:53Z2023-08-03T07:16:53Z