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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Communicated by Jens Marklof.
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Brammall, M., Winn, B. Quantum Ergodicity for Quantum Graphs without Back-Scattering. Ann. Henri Poincaré 17, 1353–1382 (2016). https://doi.org/10.1007/s00023-015-0435-8
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DOI: https://doi.org/10.1007/s00023-015-0435-8