Differential Identities for the Structure Function of Some Random Matrix Ensembles

Abstract

The structure function of a random matrix ensemble can be specified in terms of the covariance of the linear statistics ∑j=1Neik1λj, ∑j=1Ne-ik2λj for Hermitian matrices, and the same with the eigenvalues λj replaced by the eigenangles θj for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation ρ(2). For the circular β-ensemble of unitary matrices, and with β even, we characterise the bulk scaling limit of ρ(2) as the solution of a linear differential equation of order β+ 1 —a duality relates ρ(2) with β replaced by 4 / β to the same equation. Asymptotics obtained in the case β= 6 from this characterisation are combined with previous..