Differential Identities for the Structure Function of Some Random Matrix Ensembles

The structure function of a random matrix ensemble can be specified in terms of the covariance of the linear statisticse Sigma(N)(j=1), e(ik1 lambda j), Sigma(N)(j=1) e(-ik2 lambda j) for Hermitian matrices, and the same with the eigenvalues lambda j replaced by the eigenangles theta(j) for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation rho((2)). For the circular beta-ensemble of unitary matrices, and with beta even, we characterise the bulk scaling limit of rho((2)) as the solution of a linear differential equation of order beta + 1-a duality relates rho((2)) with beta replaced by 4/beta to the same equation. Asymptotics obtained in the case beta = 6 from this characterisation are combined with previously established results to determine the explicit form of the degree 10 palindromic polynomial in beta/2 which determines the coefficient of vertical bar k vertical bar 11 in the small vertical bar k vertical bar expansion of the structure function for general beta > 0. For the Gaussian unitary ensemble we give a reworking of a recent derivation and generalisation, due to Okuyama, of an identity relating the structure function to simpler quantities in the Laguerre unitary ensemble first derived in random matrix theory by Brezin and Hikami. This is used to determine various scaling limits, many of which relate to the dip-ramp-plateau effect emphasised in recent studies of many body quantum chaos, and allows too for rates of convergence to be established.