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

Research paper by **Zhigang Bao, Guangming Pan, Wang Zhou**

Indexed on: **26 Oct '13**Published on: **26 Oct '13**Published in: **Mathematics - Probability**

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#### 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.