Quantum Ergodicity for Quantum Graphs without Back-Scattering
Quantum Ergodicity for Quantum Graphs without Back-Scattering
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DOI：
10.1007/s00023-015-0435-8
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发表时间：
2016-06-01
影响因子：
1.5
通讯作者：
Winn, B.
中科院分区：
文献类型：
--
作者：
Brammall, Matthew;Winn, B.
我们给出了用禁止后向散射的边界散射矩阵量化的d-正则图的量子方差估计。对于具有很少短圈的扩展图族，我们的估计导致了这些图族的量子遍历。我们的证明是基于图的键上的相关随机游动的一致控制。我们证明了Ramanujan图和渐近必然的随机d-正则图的最新构造满足得出量子遍历性成立的必要条件。
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.