TY - JOUR
T1 - Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit
AU - Anantharaman, Nalini
AU - Ingremeau, Maxime
AU - Sabri, Mostafa
AU - Winn, Brian
N1 - Funding Information:
N.A. was supported by Institut Universitaire de France , by the ANR project GeRaSic ANR-13-BS01-0007 and by USIAS ( University of Strasbourg Institute of Advanced Study ).
Funding Information:
M.I. was supported by the Labex IRMIA during part of the redaction of this paper.
Funding Information:
M.S. was supported by a public grant as part of the Investissement d'avenir project, reference ANR-11-LABX-0056-LMH , LabEx LMH. He thanks the Université Paris Saclay for excellent working conditions, where part of this work was done.
Publisher Copyright:
© 2021
PY - 2021/6/15
Y1 - 2021/6/15
N2 - 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.
AB - 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.
KW - Benjamini-Schramm convergence
KW - Empirical spectral measures
KW - Quantum graphs
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U2 - 10.1016/j.jfa.2021.108988
DO - 10.1016/j.jfa.2021.108988
M3 - Article
AN - SCOPUS:85102847764
SN - 0022-1236
VL - 280
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 12
M1 - 108988
ER -