Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit

TL;DR

This paper introduces a new notion of convergence for quantum graphs, called Benjamini-Schramm convergence, and proves that the empirical spectral measures of such convergent sequences tend to a limiting spectral measure.

Contribution

It extends the concept of Benjamini-Schramm convergence from discrete to quantum graphs and establishes spectral measure convergence in this new setting.

Findings

Any sequence of quantum graphs with bounded data has a convergent subsequence.

Empirical spectral measures of convergent quantum graph sequences converge to a limiting spectral measure.

The proofs differ significantly from the discrete case, adapting to quantum graph complexities.

Abstract

We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a randomly chosen ball has a limiting distribution. We prove that any sequence of quantum graphs with uniformly bounded data has a convergent subsequence in this sense. We then consider the empirical spectral measure of a convergent sequence (with general boundary conditions and edge potentials) and show that it converges to the expected spectral measure of the limiting random rooted quantum graph. These results are similar to the discrete case, but the proofs are significantly different.