# Massive Modes for Quantum Graphs

## TL;DR

This paper investigates the role of massive modes in the spectral statistics of chaotic quantum graphs, demonstrating their negligible contribution under certain conditions, thus supporting the universality of spectral correlations.

## Contribution

It provides a proof that massive modes in the supersymmetry approach vanish in the large graph limit, under the assumption of a finite spectral gap.

## Key findings

- Massive modes are shown to be negligible in the large graph limit.
- The proof relies on the assumption of a finite spectral gap of the Perron-Frobenius operator.
- Supports the universality of spectral two-point functions in chaotic quantum graphs.

## Abstract

The spectral two-point function of chaotic quantum graphs is expected to be universal. Within the supersymmetry approach, a proof of that assertion amounts to showing that the contribution of non-universal (or massive) modes vanishes in the limit of infinite graph size. Here we pay particular attention to the fact that the massive modes are defined in a coset space. Using the assumption that the spectral gap of the Perron-Frobenius operator remains finite in the limit, we then argue that the massive modes are indeed negligible.