Universality for a global property of the eigenvectors of Wigner matrices

Abstract: Let $M_n$ be an $n\times n$ real (resp. complex) Wigner matrix and $U_n\Lambda_n U_n^*$ be its spectral decomposition. Set $(y_1,y_2...,y_n)^T=U_n^*x$, where $x=(x_1,x_2,...,$ $x_n)^T$ is a real (resp. complex) unit vector. Under the assumption that the elements of $M_n$ have 4 matching moments with those of GOE (resp. GUE), we show that the process $X_n(t)=\sqrt{\frac{\beta n}{2}}\sum_{i=1}^{\lfloor nt\rfloor}(|y_i|^2-\frac1n)$ converges weakly to the Brownian bridge for any $\mathbf{x}$ such that $||x||_\infty\rightarrow 0$ as $n\rightarrow \infty$, where $\beta=1$ for the real case and $\beta=2$ for the complex case. Such a result indicates that the othorgonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthorgonal (resp. unitary) group from a certain perspective.