Fluctuations of Eigenvalues of Random Hermitian Matrices

摘要

The authors consider ensembles of N x N hermitian matrices including the GaussianUnitary Ensemble. The eigenvalues behave as a log-gas on R confined by a polynomial potential V at the inverse temperature Beta = 2. For certain V the asymptotic eigenvalue distribution as N -> Infinity is supported on a single interval, and for these ensembles the authors prove a central limit theorem for the global fluctuations of the eigenvalues around the asymptotic distribution. This result is an analogue of the strong Szego limit theorem for Toeplitz determinants, which gives the global fluctuations of the eigenvalues of random unitary matrices with respect to Haar measure. As a by-product the authors analyze the asymptotics of the recursion coefficients for the orthonormal polynomials with respect to the weight exp(-NV(t)) on R. The weak convergence to the asymptotic eigenvalue distribution is proved for more general V. The authors also prove the central limit theorem for a log-gas at an arbitrary inverse temperature Beta > 0; the cases Beta = 1 and Beta = 4, V(t) = t squared correspond to the Gaussian Orthogonal Ensemble and the Gaussian Symplectic Ensemble respectively.