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The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

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Abstract

We prove that the distribution of eigenvectors of generalized Wigner 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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Authors and Affiliations

  1. Cambridge University, Cambridge, UK

    P. Bourgade

  2. Institute for Advanced Study, Princeton, USA

    P. Bourgade

  3. Harvard University, Cambridge, USA

    H.-T. Yau

Authors
  1. P. Bourgade
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  2. H.-T. Yau
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Corresponding author

Correspondence to P. Bourgade.

Additional information

Communicated by M. Salmhofer

The work of P. Bourgade is partially supported by the NSF Grant DMS1208859. The work of H.-T. Yau is partially supported by the NSF Grant DMS1307444 and the Simons Investigator Fellowship.

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Bourgade, P., Yau, HT. The Eigenvector Moment Flow and Local Quantum Unique Ergodicity. Commun. Math. Phys. 350, 231–278 (2017). https://doi.org/10.1007/s00220-016-2627-6

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  • DOI: https://doi.org/10.1007/s00220-016-2627-6

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