qe_graphs.pdf (442.94 kB)

Quantum ergodicity for quantum graphs without back-scattering

Download (442.94 kB)
journal contribution
posted on 10.09.2015 by Matthew Brammall, Brian 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.

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 Poincaré

Volume

17

Issue

6

Pages

1353 - 1382

Citation

BRAMMALL, M. and WINN, B., 2016. Quantum ergodicity for quantum graphs without back-scattering. Annales Henri Poincare. 17(6), pp.1353-1382.

Publisher

© Springer

Version

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

16/06/2015

Publication date

2015-09-29

Notes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00023-015-0435-8

ISSN

1424-0637

eISSN

1424-0661

Language

en

Exports

Logo branding

Keyword(s)

Exports