- Series
- Stochastics Seminar
- Time
- Tuesday, April 24, 2012 - 11:00am for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- F. Benaych-Georges – Universite Pierre et Marie Curie
- Organizer
- Christian Houdré
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.