Wegner estimate and level repulsion for Wigner random matrices
By L. Erdős, B. Schlein, Horng-Tzer Yau. 2008
192 citations recorded by the source corpus.
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1}$. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result \cite{ESY2}. We then show a Wegner estimate, i.e. that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.
Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/408d33254c01eedf3da4dda8d7ada0f8f2528013
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Citations in the corpus
- Tracy-Widom law for the extreme eigenvalues of sample correlation matrices (2011, unknown): “It turns out to be quite similar to the case of sample covariance matrices and Wigner matrices, see [7], [8], [18] and [20].”
- Random covariance matrices: Universality of local statistics of eigenvalues (2009, methodology): “one. In the proof of the Four Moment Theorem (as well as the Gap Theorem) for n×n Wigner matrices in [46], a crucial ingredient was a variant of the Delocalization Theorem of Erdo¨s, Schlein, and Yau [9, 10, 11]. This result asserts (assuming uniformly exponentially decaying distribution for the coe”
- Bulk universality for Wigner matrices (2009, background): “We first prove Theorem 1.2 for the ensemble Ĥ+aV with a = N−1/2+λ/2 (see the beginning of Section 3 for the necessary rescaling).”
- The local relaxation flow approach to universality of the local statistics for random matrices (2009, methodology): “As an input of this method, we needed a fairly detailed control on the local density of eigenvalues that could be obtained from our previous works on Wigner matrices [17, 18, 19].”
- A Dynamical Approach to Random Matrix Theory (2017, methodology): “Quadratic functionals were considered in [62,69]; the current formulation is taken from [52].”
- Lectures on the local semicircle law for Wigner matrices (2016, unknown): “04 05 5v 3 [ m at h. PR ] 1 8 O ct 2 01 6 Contents”
- Eigenvector distribution of Wigner matrices (2011, methodology): “We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble.”
- Universality for the largest eigenvalue of sample covariance matrices with general population (2013, background): “The delocalization property for the eigenvectors of random matrices was also raised in [12] and improved in the series work [13, 14, 19] for Wigner matrices and extended to sample covariance matrices in [35, 39, 16, 31], which asserts that the coefficients of the unit eigenvectors are in the order o”
- Localization and delocalization of eigenvectors for heavy-tailed random matrices (2012, background): “1 A geometric upper bound In this paragraph, we recall a general upper bound for NI that is due to Erdős–Schlein– Yau [20], namely if we let A(k) be the principal minor matrix of A where the k-th row and column have been removed, W (k) be the vector space generated by the eigenvectors of A(k) corres”