Ovidiu Costin, Joel L. Lebowitz
Gaussian Fluctuation in Random Matrices
(20K, TeX)

ABSTRACT.  Let $N(L)$ be the  number of eigenvalues, 
in an interval of length $L$, of a
matrix chosen at random from the 
Gaussian Orthogonal, Unitary or Symplectic ensembles of ${\cal N}$ by ${\cal N}$ matrices, in the limit  ${\cal N}\rightarrow\infty$.
We prove that $[N(L) - \langle N(L)\rangle]/\sqrt{\log L}$ has a Gaussian
distribution when $L\rightarrow\infty$.
This theorem, which 
requires control of all the higher moments of the distribution,
elucidates numerical and exact results on chaotic quantum systems and 
on the statistics of zeros of the Riemann
zeta function.