BRAMMALL, M. and WINN, B., 2016. Quantum ergodicity for quantum graphs without back-scattering. Annales Henri Poincare. 17(6), pp.1353-1382.
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.
The final publication is available at Springer via http://dx.doi.org/10.1007/s00023-015-0435-8
This paper has been supported by EPSRC under grant numbers EP/H046240/1 and EP/I038217/1.