Random matrices: Universal properties of eigenvectors
By Terence Tao, Van H. Vu. 2011
124 citations recorded by the source corpus.
The four moment theorem asserts, roughly speaking, that the joint distribution of a small number of eigenvalues of a Wigner random matrix (when measured at the scale of the mean eigenvalue spacing) depends only on the first four moments of the entries of the matrix. In this paper, we extend the four moment theorem to also cover the coefficients of the eigenvectors of a Wigner random matrix. A similar result (with different hypotheses) has been proved recently by Knowles and Yin, using a different method. As an application, we prove some central limit theorems for these eigenvectors. In another application, we prove a universality result for the resolvent, up to the real axis. This implies universality of the inverse matrix.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/2278886e3bb62d58e6ceb188ba454701fe70b215
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Eigenvector Statistics of Sparse Random Matrices (2016, background): “For Wigner type random matrices, it is proved in a series of papers [5,13–20,25,26,37] for the bulk and [21,32,36] for the edge, that the eigenvalue statistics are universal; it is proved in [6, 24, 38] that the eigenvectors are asymptotically normal.”
- Statistical Inference on Random Dot Product Graphs: a Survey (2017, background): “We stress that aside from specific cases (see [39], [102], [56]), limiting distributions for eigenvalues and eigenvectors of random graphs are notably elusive.”
- Delocalization of eigenvectors of random matrices with independent entries (2013, background): “Only recently was it proved by L. Erdös et al. that general symmetric and Hermitian random matrices H with independent entries are completely delocalized ([10, 11, 12, 14, 29], see also surveys [13, 4]).”
- On the principal components of sample covariance matrices (2014, methodology): “) We note that even this fixed-index universality of eigenvalues is a new result, having previously only been established under the four-moment matching condition [29,41] (in the context of Wigner matrices).”
- A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs (2013, background): “In particular, Knowles and Yin (2011) gives a central limit theorem for eigenvectors near the spectral edge; assuming an additional four-moment matching condition, both Tao and Vu (2012) and Knowles and Yin (2011) exhibit similar results for the bulk of the spectrum.”
- Universality of local spectral statistics of random matrices (2011, methodology): “inally, we mention that extension of universality to eigenvectors near the edge was obtained by Knowles and Yin [44] under the two moment matching condition and with four moment matching condition in [64]. Although we have focused only on Wigner matrices and β-ensembles, the ideas summarized in this”
- Eigenvector distribution of Wigner matrices (2011, unknown): “…this into (3.44), we find with high probability Eab θ [∫ I xS q(yS) dE ] − Eab θ [∫ I xR q(yR) dE ] = Eab θ ′ [∫ I xR q(yR) dE ] × [∫ I ( ∆x0 q(y R) + xRq′(yR)∆y0 +∆x0q ′(yR)∆y0 + x Rq′′(yR)(∆y0) 2 ) dE ] + 1 2 Eab θ ′′ [∫ I xR q(yR) dE ] [∫ I ( ∆x0 q(y R) + xRq′(yR)∆y0 ) dE ]2 +A4 + o(N −2+t) .”
- 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.”