Level spacings distribution for large random matrices: Gaussian fluctuations


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 $\eta_N(s)$ the number of nearest neighbors spacings greater than $s>0$ (smaller than $s>0$) we prove functional limit theorem for the process $(\eta_N(s)-\mathbb{E}\eta_N(s))/N^{1/2}$, giving weak convergence of this distribution ot some Guassian random process on $[0,\infty)$.

The limiting Gaussian random process is universal for all classical compact groups. It is Hölder continuous with any exponent less than $1/2$. Similar results can be obtained for the $n$-level-spacings distribution.


Alexander Soshnikov