Gaussian Fluctuation in Random Matrices

Details Gaussian Fluctuation in Random Matrices Gaussian Fluctuation in Random Matrices

1994-12-06

arxiv.org/abs/chao-dyn/9412004v1

Nonlinear Sciences

Chaotic Dynamics

13 pages

Scientific paper

10.1103/PhysRevLett.75.69

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. \noindent PACS nos. 05.45.+b, 03.65.-w

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