Universality of Wigner random matrices: a survey of recent results

@article{Erds2010UniversalityOW,
  title={Universality of Wigner random matrices: a survey of recent results},
  author={L{\'a}szl{\'o} Erdős},
  journal={Russian Mathematical Surveys},
  year={2010},
  volume={66},
  pages={507 - 626}
}
  • L. Erdős
  • Published 6 April 2010
  • Mathematics
  • Russian Mathematical Surveys
This is a study of the universality of spectral statistics for large random matrices. Considered are 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 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… 

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We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under

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We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e−U(x). We prove that the eigenvalue statistics in the bulk

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