Differential identities for the structure function of some random matrix ensembles

The structure function of a random matrix ensemble can be specified as the covariance of the linear statistics $\sum_{j=1}^N e^{i k_1 \lambda_j}$, $\sum_{j=1}^N e^{-i k_2 \lambda_j}$ 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 $|k|^{11}$ in the small $|k|$ 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 Br\'ezin 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.