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

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.

38 pages

References in corpus (3)

• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• The Canopy Graph and Level Statistics for Random Operators on Trees
• An Invitation to Random Schroedinger operators

Cited by in corpus (1)

• Length spectrum of large genus random metric maps