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 $U_n{\Lambda }_n{U}_n^{\ast }$ be its spectral decomposition. Set $(y_1,y_2...,y_n{)}^T=U_n^{\ast }x$, 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 $X_n(t)=\sqrt{\frac{\beta n}{2}}{\sum }_{i=1}^{\lfloor nt\rfloor }(|y_i{|}^2-\frac{1}{n})$ converges weakly to the Brownian bridge for any $\mathbf{x}$ such that $||x|{|}_{\infty }\to 0$ as $n\to \infty$, 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.

typos corrected

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Random matrices: Universality of local eigenvalue statistics up to the edge