Gaussian fluctuation in random matrices.

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 N by N matrices, in the limit N → ∞. We prove that [N(L) − 〈N(L)〉]/√logL has a Gaussian distribution when L → ∞. 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. PACS nos. 05.45.+b, 03.65.-w 1 Ensembles of N dimensional Gaussian Random Matrices (GRM) with invariances under the Orthogonal, Unitary or Symplectic groups corresponding to the GOE, GUE and GSE were introduced by Wigner and developed by Porter, Dyson, Mehta and others [1,2]. Wigner’s inspired surmise that the statistics of eigenvalues of these GRM can be used to model the statistical properties of the observed spectra of complex nuclei turned out to be exactly right. There is indeed good agreement between the observed high energy level spacings, pair correlations and variance or ∆-statistics and those calculated analytically from the GRM in the limit N → ∞. Moreover, the GRM have been found to be the very robust “renormalization group fixed points” of a large class of RM [3] which play an important role in many areas of physics and mathematics [1–7]. In the present work we focus on the large L (long wavelength) behavior of the random variableN(L) giving the number of eigenvalues of a GRM, chosen from any of the Gaussian ensembles, in an interval (y, y+L): We always consider theN → ∞ limit when the distribution is translation invariant and use units in which the mean spacing is unity. It is well known that the variance of N(L) grows like logL as L → ∞. We prove that all the moments of ξ(L) ≡ [N(L)− L]/√logL approach for large L those of a Gaussian distribution which implies (weak) convergence of ξ(L) to a Gaussian random variable. We shall discuss later the connection of our result with the statistics of energy levels of quantum systems with generic chaotic 2 classical Hamiltonians and of the zeros of the Riemann zeta function [4,5,7]. It is a remarkable fact that the distribution of energy levels of the G(O,U,S)E are given by the Gibbs canonical distribution of the positions of charged point particles on the line interacting via the (two dimensional) repulsive logarithmic Coulomb potential, v(r) = − log r, at reciprocal temperatures β = 1, 2, 4 respectively [1,2]. The particles with positions xi, i = 1, ...,N , on the real line, are confined by a uniform negative background, which produces a harmonic potential. The total potential energy of the system is VG(x1, ..., xN ) = 1 2 N