Dyson’s correlation functions and graded symmetry

Abstract

A new derivation of Dyson’s k-level correlation functions of the Gaussian unitary ensemble (GUE) is given. The method uses matrices with graded symmetry. The number of integrations needed for the ensemble average becomes independent of the level number N. For arbitrary level number N, the k-level correlation function is expressed as an integral involving the eigenvalues of a 2k×2k graded matrix. The limit of infinitely many levels N→∞ is calculated by a simple saddle-point approximation of this integral, avoiding the introduction of Hermite polynomials and oscillator wave functions.