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Article Dans Une Revue Random Matrices: Theory and Applications Année : 2012

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

Florent Benaych-Georges
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Résumé

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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Dates et versions

hal-00583889 , version 1 (07-04-2011)
hal-00583889 , version 2 (22-04-2011)
hal-00583889 , version 3 (27-04-2011)
hal-00583889 , version 4 (30-05-2011)
hal-00583889 , version 5 (15-06-2011)
hal-00583889 , version 6 (28-09-2012)

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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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