Quantum Ergodicity for Quantum Graphs without Back-Scattering |
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Authors: | Matthew Brammall B Winn |
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Institution: | 1.Helix Building, Kelvin Campus,West of Scotland Science Pk.,Glasgow,United Kingdom;2.Department of Mathematical Sciences,Loughborough University,Loughborough,United Kingdom |
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Abstract: | 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. |
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