  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.
