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

Abstract

The ensemble average of $|{\sum }_{j=1}^N{e}^{ik{\lambda }_j}{|}^2$ 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 Brézin 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 $\frac{2}{\pi }\mathrm{Arctan}k$, and so there is no ramp-plateau transition.

23 pages; v2 reference added, Introduction updated; v3 incorporates corrections and suggestions of the referees

References in corpus (4)

• Chaos, Complexity, and Random Matrices
• Complete random matrix classification of SYK models with $\mathcal{N}=0$, $1$ and $2$ supersymmetry
• Spectral decoupling in many-body quantum chaos
• Differential identities for the structure function of some random matrix ensembles

Cited by in corpus (5)

• On the Spectral Form Factor for Random Matrices
• Dip-ramp-plateau for Dyson Brownian motion from the identity on $U(N)$
• No-resonance conditions, random matrices, and quantum chaotic models
• Statistics of the Random Matrix Spectral Form Factor
• Anatomy of information scrambling and decoherence in the integrable Sachdev-Ye-Kitaev model