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Quantum ergodicity for quantum graphs without back-scattering
journal contributionposted on 2015-09-10, 13:36 authored by Matthew Brammall, Brian WinnBrian Winn
We give an estimate of the quantum variance for d-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random d-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.
This paper has been supported by EPSRC under grant numbers EP/H046240/1 and EP/I038217/1.
- Mathematical Sciences
Published inAnnales Henri Poincaré
Pages1353 - 1382
CitationBRAMMALL, M. and WINN, B., 2016. Quantum ergodicity for quantum graphs without back-scattering. Annales Henri Poincare. 17(6), pp.1353-1382.
- AM (Accepted Manuscript)
Publisher statementThis work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
NotesThe final publication is available at Springer via http://dx.doi.org/10.1007/s00023-015-0435-8