### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-48 |

Number of pages | 48 |

Journal | Communications in Mathematical Physics |

DOIs | |

State | Accepted/In press - Apr 30 2016 |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*, 1-48. https://doi.org/10.1007/s00220-016-2627-6

**The Eigenvector Moment Flow and Local Quantum Unique Ergodicity.** / Bourgade, Paul; Yau, H. T.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - The Eigenvector Moment Flow and Local Quantum Unique Ergodicity

AU - Bourgade, Paul

AU - Yau, H. T.

PY - 2016/4/30

Y1 - 2016/4/30

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84964691552&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964691552&partnerID=8YFLogxK

U2 - 10.1007/s00220-016-2627-6

DO - 10.1007/s00220-016-2627-6

M3 - Article

SP - 1

EP - 48

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

ER -