Universality for a global property of the eigenvectors of Wigner matrices

Let Mn be an n×n real (resp. complex) Wigner matrix and [12pt]{minimal}{document}UₙΛ ₙ Uₙ^*{document}UnΛnUn* be its spectral decomposition. Set [12pt]{minimal}{document}(y₁,y₂⋯ ,yₙ)T=Uₙ^*x{document}(y1,y2⋯,yn)T=Un*x, where x = (x1, x2, ⋅⋅⋅, xn)T is a real (resp. complex) unit vector. Under the assumption that the elements of Mn have 4 matching moments with those of GOE (resp. GUE), we show that the process [12pt]{minimal}{document}Xₙ(t)=√β n/2∑ ᵢ₌₁ⁿᵗ(|yᵢ|²-1/n){document}Xn(t)=βn2∑i=1⌊nt⌋(|yi|2−1n) converges weakly to the Brownian bridge for any x satisfying ‖x‖∞ → 0 as n → ∞, where β = 1 for the real case and β = 2 for the complex case. Such a result indicates that the orthogonal (resp. unitary) matrices with columns being the eigenvectors of Wigner matrices are asymptotically Haar distributed on the orthogonal (resp. unitary) group from a certain perspective.