The structure function of a random matrix ensemble can be specified in terms of 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 Brézin 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.

可以根据线性统计量\（\ sum _ {j = 1} ^ N e ^ {i k_1 \ lambda _j} \），\（\ sum _ {对于Hermitian矩阵，j = 1} ^ N e ^ {-i k_2 \ lambda _j} \），对于unit矩阵，特征值\（\ lambda _j \）替换为特征角\（\ theta _j \）相同。因此，可以根据密度-密度相关性\（\ rho _ {（2）} \）的傅立叶变换来写。对于unit矩阵的圆\（\ beta \）集合，甚至使用\（\ beta \），我们表征\（\ rho _ {（2）} \）的整体缩放极限作为阶数\（\ beta + 1 \）的线性微分方程的解—对偶关系\（\ rho _ {（2）} \）由\（\ beta \）替换为\（4 / \ beta \ ）到相同的方程式。从该特征获得的情况下，在\（\ beta = 6 \）的渐近性与先前建立的结果相结合，以确定\（\ beta / 2 \）的10次​​回文多项式的显式形式，该形式决定了\（| k | ^ {11} \）中的小| k | 一般\（\ beta> 0 \）的结构函数的扩展。对于高斯一元系综，由于奥山（Okuyama）的缘故，我们对最近的推导和归纳进行了重制，该身份与结构函数与Laguerre一元系综中的简单量有关，该关系首先由Brézin和Hikami在随机矩阵理论中得出。这用于确定各种缩放比例限制，其中许多与最近在许多体量子混沌研究中强调的倾斜-斜坡-平稳效应有关，并且也允许建立收敛速率。