Alexander Soshnikov
Global Level Spacings Distribution for Large Random Matrices :
Gaussian Fluctuations
(340K, ps)
ABSTRACT. We study the level-spacings distribution for eigenvalues of large
$ \ N \times N \ $ matrices from the Classical Compact Groups in the
scaling limit when the mean distance between nearest eigenvalues
equals 1.\\
Defining by $\ \eta_N(s) \ $ the number of nearest neighbors spacings,
greater tnan $ \ s>0 \ $ (smaller than $ \ s>0 \ $ ) we prove functional
limit theorem for the process
\\ $ (\eta_N(s)-E \eta_N(s))/N^{1/2} $,
giving weak convergence of this distribution to some Gaussian random
process on $ \ \ \ [0, \infty ) \ \ $.\\
The limiting Gaussian random process is universal for all Classical
Compact Groups. It is H\"older continuous with any exponent less
than $ \ \ 1/2 \ \ .$
Numerical results suggest it not to be a standard Brownian bridge.\\
Our methods can be also applied to study n-level spacings distribution.