Departure of some parameter-dependent spectral statistics of irregular quantum graphs from Random Matrix Theory predictions

TL;DR

This paper investigates how the spectral statistics of irregular quantum graphs deviate from Random Matrix Theory predictions, focusing on parameter-dependent properties, level velocities, curvatures, and wave function localization.

Contribution

It provides a detailed comparison between spectral statistics of irregular quantum graphs and RMT, incorporating localization analysis via IPR and amplitude distributions.

Findings

Spectral statistics show deviations from RMT predictions.

Level velocities and curvatures are characterized and compared.

Localization phenomena are analyzed using IPR and amplitude distributions.

Abstract

Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) and c(w,x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of Random Matrix Theory (RMT) for Gaussian Orthogonal Ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Psi(x) using the Inverse Participation Ratio (IPR) and the amplitude distribution P(Psi(x)).