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Quantum ergodicity for quantum graphs without back-scattering
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.
Funding
This paper has been supported by EPSRC under grant numbers EP/H046240/1 and EP/I038217/1.
History
School
- Science
Department
- Mathematical Sciences
Published in
Annales Henri PoincaréVolume
17Issue
6Pages
1353 - 1382Citation
BRAMMALL, M. and WINN, B., 2016. Quantum ergodicity for quantum graphs without back-scattering. Annales Henri Poincare. 17(6), pp.1353-1382.Publisher
© SpringerVersion
- AM (Accepted Manuscript)
Publisher statement
This 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/Acceptance date
2015-06-16Publication date
2015-09-29Notes
The final publication is available at Springer via http://dx.doi.org/10.1007/s00023-015-0435-8ISSN
1424-0637eISSN
1424-0661Publisher version
Language
- en