The Eigenvector Moment Flow and Local Quantum Unique Ergodicity
By P. Bourgade, Horng-Tzer Yau. 2013
145 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/858f179487a66d0bd4b77ff2d11bf4c6dee7f820
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In Discrete Mathematics and Combinatorics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- A Dynamical Approach to Random Matrix Theory (2017, result): “2 of [28] (a similar result was also obtained in [87, 132] under the condition that the first four moments match with those of the Gaussian): Theorem 18.”
- Cleaning large correlation matrices: tools from random matrix theory (2016, methodology): “This is the approach used in [125,126,174] in order to study the mean squared overlap (4.”
- Eigenvector Statistics of Sparse Random Matrices (2016, methodology): “With it as input, combining with the rigidity estimates of eigenvalues from [26], we analyze the eigenvector moment flow of Dyson Brownian motion following the approach developed by the first and last author in [6].”
- On the principal components of sample covariance matrices (2014, methodology): “Finally, the eigenvectors of Q are uniformly distributed on the unit sphere of R ; following [15], we call this property the quantum unique ergodicity of the eigenvectors of Q, a term borrowed from quantum chaos.”
- Edge universality for deformed Wigner matrices (2014, methodology): “) The continuity of the DBM was used in [6] to compare the local eigenvalue statistics along the flow of the DBM for very short times.”