Quantifying Dip–Ramp–Plateau for the Laguerre Unitary Ensemble Structure Function

Abstract

The ensemble average of $| \sum_{j=1}^N e^{i k \lambda_j} |^2$ is of interest as a probe of quantum chaos, as is its connected part, the structure function. Plotting this average for model systems of chaotic spectra reveals what has been termed a dip-ramp-plateau shape. Generalising earlier work of Br\'ezin and Hikami for the Gaussian unitary ensemble, it is shown how the average in the case of the Laguerre unitary ensemble can be reduced to an expression involving the spectral density of the Jacobi unitary ensemble. This facilitates studying the large $N$ limit, and so quantifying the dip-ramp-plateau effect. When the parameter $a$ in the Laguerre weight $x^a e^{-x}$ scales with $N$, quantitative agreement is found with the characteristic features of this effect known for the Gaussian unitary ensemble. However, for the parameter $a$ fixed, the bulk scaled structure function is shown to have the simple functional form ${2 \over \pi} {\rm Arctan} \, k$, and so there is no ramp-plateau transition.