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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...
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| Format: | Default Article |
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2015
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| Online Access: | https://hdl.handle.net/2134/18694 |
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| _version_ | 1818172689189175296 |
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| author | Matthew Brammall Brian Winn |
| author_facet | Matthew Brammall Brian Winn |
| author_sort | Matthew Brammall (7162037) |
| collection | Figshare |
| description | 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. |
| format | Default Article |
| id | rr-article-9387605 |
| institution | Loughborough University |
| publishDate | 2015 |
| record_format | Figshare |
| spelling | rr-article-93876052015-09-29T00:00:00Z Quantum ergodicity for quantum graphs without back-scattering Matthew Brammall (7162037) Brian Winn (1247334) Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified 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. 2015-09-29T00:00:00Z Text Journal contribution 2134/18694 https://figshare.com/articles/journal_contribution/Quantum_ergodicity_for_quantum_graphs_without_back-scattering/9387605 CC BY-NC-ND 4.0 |
| spellingShingle | Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified Matthew Brammall Brian Winn Quantum ergodicity for quantum graphs without back-scattering |
| title | Quantum ergodicity for quantum graphs without back-scattering |
| title_full | Quantum ergodicity for quantum graphs without back-scattering |
| title_fullStr | Quantum ergodicity for quantum graphs without back-scattering |
| title_full_unstemmed | Quantum ergodicity for quantum graphs without back-scattering |
| title_short | Quantum ergodicity for quantum graphs without back-scattering |
| title_sort | quantum ergodicity for quantum graphs without back-scattering |
| topic | Other mathematical sciences not elsewhere classified untagged Mathematical Sciences not elsewhere classified |
| url | https://hdl.handle.net/2134/18694 |