Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
By L'aszl'o ErdHos, B. Schlein, Horng-Tzer Yau. 2007
249 citations recorded by the source corpus.
We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. We study the connection between eigenvalue statistics on microscopic energy scales $\eta\ll1$ and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order $\eta \sim\log N/N$. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales $\eta\gg N^{-2/3}$. We show that most eigenvectors are fully delocalized in the sense that their $\ell^p$-norms are comparable with $N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound $N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.
Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/d58c8be4c71ffc6a5cc0d66a6fab0b3736054869
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Citations in the corpus
- 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”
- Eigenvector distribution of Wigner matrices (2011, background): “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.”
- Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues (2011, methodology): “Neglecting the mean zero condition, the Erdős-Rényi matrix becomes a Wigner random matrix with a Bernoulli distribution when 0 p 1 is a constant independent of N .”
- Delocalization of eigenvectors of random matrices with independent entries (2013, methodology): “that general symmetric and Hermitian random matrices H with independent entries are completely delocalized ([10, 11, 12, 14, 29], see also surveys [13, 4]).”
- Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, background): “The arguments are well established in the literature; indeed, the z = 0 case of these theorems essentially appeared in [57], [21], while the analogous estimates for Wigner matrices appeared in [17], [18], [19], [56].”
- Universality of local spectral statistics of random matrices (2011, methodology): “This theorem is the final result of subsequent improvements in [31, 32, 36, 37, 38] of our first local semicirle law in [30].”
- Extremal eigenvalues and eigenvectors of deformed Wigner matrices (2013, methodology): “[23] Erdős, L., Yau, H.-T., Yin, J.: Bulk Universality for Generalized Wigner Matrices, Probab.”
- Rigidity of eigenvalues of generalized Wigner matrices (2010, background): “In a series of papers [16, 17, 18, 20, 19, 21, 22, 23], we developed a new approach to understand local eigenvalue statistics.”
- A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “) For general Wigner matrices, a new approach was introduced to prove the bulk universality in a series of papers by Erdős, Schlein, Yau, and others in [17, 18, 19, 20, 21, 24, 25, 26, 15, 16].”
- Random matrices: Sharp concentration of eigenvalues (2012, background): “bjects, suchastheStieltjes transformsWn(z) := 1 n trace(Wn− z)−1 of W n) for short intervals I, due to the applicability of such results to establishingvariousuniversalitypropertiesofsuchmatrices; see[11, 12, 13, 30,31, 14, 16,17]. The previous best result in this direction was by Erdo˝s, Yau, and Y”
- A Dynamical Approach to Random Matrix Theory (2017, background): “Since vanishing fluctuations can be used to estimate the density, this argument can actually be made rigorous by a bootstrapping argument [61].”
- RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE (2011, methodology): “Recall in the proofs of the Four Moment Theorem and the Gap Theorem as in [10], [13] and [12], a crucial input was the Delocalization Theorem of Erdős, Schlein, and Yau ([7], [6] and [5]).”