Quantifying dip-ramp-plateau for the Laguerre unitary ensemble structure function

TL;DR

This paper analyzes the dip-ramp-plateau structure function in the Laguerre unitary ensemble, extending previous work on Gaussian ensembles, and finds how the structure varies with parameters, revealing no transition in certain cases.

Contribution

It generalizes the analysis of the structure function to the Laguerre ensemble, connecting it to the Jacobi ensemble and exploring the large N limit.

Findings

For large N, the structure function matches Gaussian ensemble features when parameter a scales with N.

When a is fixed, the structure function simplifies to an arctangent form, showing no dip-ramp-plateau transition.

The analysis provides a quantitative understanding of spectral correlations in chaotic quantum systems.

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.