 94357 Ovidiu Costin, Joel L. Lebowitz
 Gaussian Fluctuation in Random Matrices
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Nov 18, 94

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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.
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