Universality of Wigner Random Matrices - Mathematical PhysicsReport as inadecuate

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Abstract: We consider $N\times N$ symmetric or hermitian random matrices withindependent, identically distributed entries where the probability distributionfor each matrix element is given by a measure $ u$ with a subexponentialdecay. We prove that the local eigenvalue statistics in the bulk of thespectrum for these matrices coincide with those of the Gaussian OrthogonalEnsemble GOE and the Gaussian Unitary Ensemble GUE, respectively, in thelimit $N\to \infty$. Our approach is based on the study of the Dyson Brownianmotion via a related new dynamics, the local relaxation flow. We also show thatthe Wigner semicircle law holds locally on the smallest possible scales and weprove that eigenvectors are fully delocalized and eigenvalues repel each otheron arbitrarily small scales.

Author: Laszlo Erdos

Source: https://arxiv.org/

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