Universality of the global fluctuations for the eigenvectors of Wigner random matrices

Stochastics Seminar
Tuesday, April 24, 2012 - 11:00
1 hour (actually 50 minutes)
Skiles 005
Universite Pierre et Marie Curie
Many of the asymptotic spectral characteristics of a symmetric random matrix with i.i.d. entries (such a matrix is called a "Wigner matrix") are said to be "universal": they depend on the exact distribution of the entries only via its first moments (in the same way that the CLT gives the asymptotic fluctuations of the empirical mean of i.i.d. variables as a function of their second moment only). For example, the empirical spectral law of the eigenvalues of a Wigner matrix converges to the semi-circle law if the entries have variance 1, and the extreme eigenvalues converge to -2 and 2 if the entries have a finite fourth moment. This talk will be devoted to a "universality result" for the eigenvectors of such a matrix. We shall prove that the asymptotic global fluctuations of these eigenvectors depend essentially on the moments with orders 1, 2 and 4 of the entries of the Wigner matrix, the third moment having surprisingly no influence.