Linear Functionals of Eigenvalues of Random Matrices

LetMn be a random n n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n ! 1. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coe cients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristicpolynomial ofMn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.