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The structure function of a random matrix ensemble can be specified as thecovariance 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 withthe eigenvalues $\lambda_j$ replaced by the eigenangles $\theta_j$ for unitarymatrices. As such it can be written in terms of the Fourier transform of thedensity-density correlation $\rho_{(2)}$. For the circular $\beta$-ensemble ofunitary matrices, and with $\beta$ even, we characterise the bulk scaling limitof $\rho_{(2)}$ 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 todetermine 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 Gaussianunitary ensemble we give a reworking of a recent derivation and generalisation,due to Okuyama, of an identity relating the structure function to simplerquantities in the Laguerre unitary ensemble first derived in random matrixtheory by Br\'ezin and Hikami. This is used to determine various scalinglimits, many of which relate to the dip-ramp-plateau effect emphasised inrecent studies of many body quantum chaos, and allows too for rates ofconvergence to be established.

Differential identities for the structure function of some random matrix ensembles