Quantum ergodicity for quantum graphs without back-scattering
journal contributionposted on 10.09.2015 by Matthew Brammall, Brian Winn
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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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This paper has been supported by EPSRC under grant numbers EP/H046240/1 and EP/I038217/1.
- Mathematical Sciences