Rigidity and normal modes in random matrix spectra

We consider the Gaussian ensembles of random matrices and describe the norm al modes of the eigenvalue spectrum, i.e., the correlated fluctuations of e igenvalues about their most probable values. The associated normal mode spe ctrum is linear, and for large matrices, the normal modes are found to be C hebyshev polynomials of the second kind. We contrast this with the behaviou r of a sequence of uncorrelated levels, which has a quadratic normal mode s pectrum. The difference in the rigidity of random matrix spectra and sequen ces of uncorrelated levels can be attributed to this difference in the norm al mode spectra. We illustrate this by calculating the number variance in t he two cases. (C) 1999 Elsevier Science B.V.