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 linear statistics $\sum_{j=1}^N e^{i k_1 \lambda_j}$, e^{-i k_2 \lambda_j}$ for Hermitian matrices, and same with eigenvalues $\lambda_j$ replaced by eigenangles $\theta_j$ unitary matrices. As such it written in terms Fourier transform density-density correlation $\rho_{(2)}$. For circular $\beta$-ensemble $\beta$ even, we characterise bulk scaling limit $\rho_{(2)}$ solution differential equation order $\beta + 1$ -- duality relates $4/\beta$ to equation. Asymptotics obtained case = 6$ from this characterisation are combined previously established results determine explicit form degree 10 palindromic polynomial $\beta/2$ which determines coefficient $|k|^{11}$ small $|k|$ expansion general > 0$. Gaussian give reworking recent derivation generalisation, due Okuyama, an identity relating simpler quantities Laguerre first derived theory Br\'ezin Hikami. This is used various limits, many relate dip-ramp-plateau effect emphasised studies body quantum chaos, allows too rates convergence established.