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**Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices.**
*(English)*
Zbl 1175.15028

E. Wigner has introduced random matrices to model Hamiltonians, \(H\), of atomic nuclei. Random Hamiltonians are also used in solid state physics to study electrons in disordered metallic lattices. Random matrices are mostly studied from the point of view of eigenvalue of statistics. In this paper the authors prove several results in the directions for general Hermitian Wigner matrices. In Theorem 2.1 they give an upper bound on the eigenvalue density. Theorem 3.1 states that the density of states concentrates around its mean in probability sense down to energy windows of order \(\gg\) power \((-2/3)\) of \(N\). In Theorem 4.1 they prove that the expectation value of the density of the states on scales \(\gg\) power \((-2/3)\) of \(N\) converges to the Wigner semicircle law. In Theorem 5.1 they show that most eigenvectors are fully extended in the sense that their \(l\)-norm is of order power \((-1/2)\) of \(N\). In Theorem 6.1 by using the bounds on the eigenvectors, they give an estimate on the second moment of the Green function. In Theorem 7.1 they prove that no eigenvector is strongly localized in the sense that no eigenvector can be essentially supported on a small percentage of the sites.

Reviewer: Yueh-er Kuo (Knoxville)

### MSC:

15B52 | Random matrices (algebraic aspects) |

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

### Keywords:

semicircle law; Wigner random matrix; random Schrödinger operator; eigenvalue bounds; density of states; localization; extended states; Hamiltonians; eigenvector
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\textit{L. Erdős} et al., Ann. Probab. 37, No. 3, 815--852 (2009; Zbl 1175.15028)

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