Spectral statistics of preferred orientation quantum graphs

TL;DR

This paper investigates how preferred orientation vertex conditions in quantum graphs affect their spectral statistics, revealing deviations from random matrix theory predictions and analyzing the underlying combinatorics of periodic orbits.

Contribution

It identifies and explains deviations in spectral statistics caused by preferred orientations, including for standard Neumann-Kirchhoff conditions, with detailed combinatorial analysis.

Findings

Deviations from random matrix theory in spectral statistics due to preferred orientations.

Discrepancies also occur in graphs with standard Neumann-Kirchhoff conditions.

Combinatorial analysis of periodic orbits explains these phenomena.

Abstract

We study the spectral statistics of quantum (metric) graphs whose vertices are equipped with preferred orientation vertex conditions. When comparing their spectral statistics to those predicted by suitable random matrix theory ensembles, one encounters some deviations. We point out these discrepancies and demonstrate that they occur in various graphs and even for Neumann-Kirchhoff vertex conditions, which was overlooked so far. Detailed explanations and computations are provided for this phenomena. To achieve this, we explore the combinatorics of periodic orbits, with a particular emphasis on counting Eulerian cycles.