Universality of Wigner random matrices: A survey of recent results

L. Erdös, Russian Mathematical Surveys 66 (2011) 507–626.

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Abstract
This is a study of the universality of spectral statistics for large random matrices. Considered are N×N symmetric, Hermitian, or quaternion self-dual random matrices with independent identically distributed entries (Wigner matrices), where the probability distribution of each matrix element is given by a measure v with zero expectation and with subexponential decay. The main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), and the Gaussian Symplectic Ensemble (GSE), respectively, in the limit as N → ∞. This approach is based on a study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, it is established that the density of the eigenvalues converges to the Wigner semicircle law, and this holds even down to the smallest possible scale. Moreover, it is shown that the eigenvectors are completely delocalized. These results hold even without the condition that the matrix elements are identically distributed: only independence is used. In fact, for the matrix elements of the Green function strong estimates are given that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. A Wigner-type estimate is also proved, and it is shown that the eigenvalues repel each other on arbitrarily small scales.
Publishing Year
Date Published
2011-01-01
Journal Title
Russian Mathematical Surveys
Volume
66
Issue
3
Page
507 - 626
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Erdös L. Universality of Wigner random matrices: A survey of recent results. Russian Mathematical Surveys. 2011;66(3):507-626. doi:10.1070/RM2011v066n03ABEH004749
Erdös, L. (2011). Universality of Wigner random matrices: A survey of recent results. Russian Mathematical Surveys. IOP Publishing Ltd. https://doi.org/10.1070/RM2011v066n03ABEH004749
Erdös, László. “Universality of Wigner Random Matrices: A Survey of Recent Results.” Russian Mathematical Surveys. IOP Publishing Ltd., 2011. https://doi.org/10.1070/RM2011v066n03ABEH004749.
L. Erdös, “Universality of Wigner random matrices: A survey of recent results,” Russian Mathematical Surveys, vol. 66, no. 3. IOP Publishing Ltd., pp. 507–626, 2011.
Erdös L. 2011. Universality of Wigner random matrices: A survey of recent results. Russian Mathematical Surveys. 66(3), 507–626.
Erdös, László. “Universality of Wigner Random Matrices: A Survey of Recent Results.” Russian Mathematical Surveys, vol. 66, no. 3, IOP Publishing Ltd., 2011, pp. 507–626, doi:10.1070/RM2011v066n03ABEH004749.

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