Spectral form factor in a random matrix theory

TL;DR

The paper investigates the spectral form factor $S(\tau)$ for the Gaussian unitary ensemble, deriving exact finite-$N$ contour-integral expressions for the two-level correlation function and revealing a universal linear regime $S(\tau) \propto \tau$ for small to moderate $\tau$ followed by nonuniversal crossovers near $\tau_c=2N$; oscillatory finite-$N corrections and energy-averaged behavior are analyzed, with a surprising link to the Laguerre ensemble via $\langle S(\tau)\rangle = -\frac{e^{-x/N}}{N^2} [\frac{d}{dx}L_N(x/N)]^2$ and universal zero-energy oscillations described by Bessel functions. The authors extend the framework to the time-dependent case, mapping to a two-matrix model with coupling $c=e^{-t}$, which smears the Heisenberg singularity and reveals nonuniversality in the two-matrix setting; they also study correlation functions with external sources, establishing kernel-based universal short-distance behavior and detailing zeros of the kernel $K_N(\lambda,\mu)$. Overall, the work provides a powerful contour-integral method to characterize universal and nonuniversal aspects of spectral statistics, including higher-point functions and zero structures, across static, time-dependent, and externally forced random-matrix ensembles.

Abstract

In the theory of disordered systems the spectral form factor $S(τ)$, the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for $τ<τ_c$ and constant for $τ>τ_c$. Near zero and near $τ_c$ its exhibits oscillations which have been discussed in several recent papers. In the problems of mesoscopic fluctuations and quantum chaos a comparison is often made with random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscilllations have not yet been studied there. For random matrices, the two-level correlation function $ρ(λ_1,λ_2)$ exhibits several well-known universal properties in the large N limit. Its Fourier transform is linear as a consequence of the short distance universality of $ρ(λ_1,λ_2)$. However the cross-over near zero and $τ_c$ requires to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these cross-over oscillatory properties. The method is also extended to the time-dependent case.