A universality result for the global fluctuations of the eigenvectors of Wigner matrices

We prove that for [ui,j ] n i,j=1 the eigenvectors matrix of a Wigner matrix, under some moments conditions, the bivariate random process BB@ X 1≤i≤ns, 1≤j≤nt (|ui,j | − 1/n) CCA (s,t)∈[0,1]2 converges in distribution to a bivariate Brownian bridge. This result has already been proved for GOE and GUE matrices. It is conjectured here that the necessary and sufficient condition, for the result to be true for a general Wigner matrix, is the matching of the moments of orders 1, 2 and 4 of the entries of the Wigner with the ones of a GOE or GUE matrix. Surprisingly, the third moment of the entries of the Wigner matrix has no influence on the limit distribution.