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

Abstract : Let $U_n=[u_{i,j}]$ be the eigenvectors matrix of a Wigner matrix. We prove that under some moments conditions, the bivariate random process indexed by $[0,1]^2$ with value at $(s,t)$ equal to the sum, over $1\le i \le ns$ and $1\le j \le nt$, of $|u_{i,j}|^2 - 1/n$, converges in distribution to the 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.
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Article dans une revue
Random Matrices: Theory and Applications, 2012, 01 (04), pp.23. 〈10.1142/S2010326312500116〉
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https://hal.archives-ouvertes.fr/hal-00583889
Contributeur : Florent Benaych-Georges <>
Soumis le : vendredi 28 septembre 2012 - 08:54:20
Dernière modification le : jeudi 31 mai 2018 - 09:12:01
Document(s) archivé(s) le : vendredi 16 décembre 2016 - 17:49:38

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Florent Benaych-Georges. A universality result for the global fluctuations of the eigenvectors of Wigner matrices. Random Matrices: Theory and Applications, 2012, 01 (04), pp.23. 〈10.1142/S2010326312500116〉. 〈hal-00583889v6〉

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