# Universality for a global property of the eigenvectors of Wigner matrices

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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 $X nt=\sqrt{\frac{\beta n}{2}}\sum {i=1}^{\lfloor nt floor}|y i|^2-\frac1n$ converges weakly to the Brownian bridge for any $\mathbf{x}$ such that $||x|| \infty ightarrow 0$ as $n ightarrow \infty$, 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.

Autor: Zhigang Bao; Guangming Pan; Wang Zhou

Fuente: https://archive.org/