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

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posted on 12.04.2021, 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.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Journal of Functional Analysis

Volume

280

Issue

12

Publisher

Elsevier BV

Version

AM (Accepted Manuscript)

Rights holder

© Crown

Publisher statement

This 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

Acceptance date

21/02/2021

Publication date

2021-03-17

Copyright date

2021

ISSN

0022-1236

Language

en

Depositor

Dr Brian Winn. Deposit date: 8 April 2021

Article number

108988

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