Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
journal contributionposted on 2021-04-12, 12:24 authored by Nalini Anantharaman, Maxime Ingremeau, Mostafa Sabri, Brian WinnBrian Winn
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.
- Mathematical Sciences
Published inJournal of Functional Analysis
- AM (Accepted Manuscript)
Rights holder© Crown
Publisher statementThis paper was accepted for publication in the journal Journal of Functional Analysis and the definitive published version is available at https://doi.org/10.1016/j.jfa.2021.108988