Level Spacings Distribution for Large Random Matrices: Gaussian Fluctuations

We study the level-spacings distribution for eigenvalues of large N X N matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1. Defining by 77N(S) the number of nearest neighbors spacings greater than s > 0 (smaller than s > 0) we prove functional limit theorem for the process (77N(S)-E%/N(S))/N1/2, giving weak convergence of this distribution to some Gaussian random process on [0, x). The limiting Gaussian random process is universal for all classical compact groups. It is Holder continuous with any exponent less than 1/2. Similar results can be obtained for the n-level-spacings distribution.