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Download fileEmpirical spectral measures of quantum graphs in the Benjamini-Schramm limit
journal contribution
posted on 2021-04-12, 12:24 authored by Nalini Anantharaman, Maxime Ingremeau, Mostafa Sabri, Brian WinnBrian WinnWe 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.
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