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The eigenvector moment flow and local quantum unique ergodicity

Bourgade P Cambridge University HT Yau Harvard University

Symplectic Geometry mathscidoc:1707.34001

Distinguished Paper Award in 2017

Commun. Math. Phys., 350, 231-278, 2017
We prove that the distribution of eigenvectors of generalizedWigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.
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@inproceedings{bourgade2017the,
  title={The eigenvector moment flow and local quantum unique ergodicity},
  author={Bourgade P, and HT Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170719190234022204801},
  booktitle={Commun. Math. Phys.},
  volume={350},
  pages={231-278},
  year={2017},
}
Bourgade P, and HT Yau. The eigenvector moment flow and local quantum unique ergodicity. 2017. Vol. 350. In Commun. Math. Phys.. pp.231-278. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170719190234022204801.
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