Power spectrum of the circular unitary ensemble

Abstract

We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble CUE(N) and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlevé function. In the limit of infinite-dimensional matrices, N→∞, we derive a concise parameter-free formula for the power spectrum which involves a fifth Painlevé transcendent and interpret it in terms of the Sine2 determinantal random point field. Further, we discuss a universality of the predicted power spectrum law and tabulate it for easy use by random-matrix-theory and quantum chaos practitioners.