Random matrices: universality of local eigenvalue statistics. (English) Zbl 1217.15043
The authors consider the universality of the local eigenvalue statistics of random matrices. The main result is Theorem 15, called the Four moment theorem. It shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence they derive the universality of the eigenvalue gap distribution and \(k\)-point correlation, and many other statistics for both Wigner Hermitian matrices and Wigner real symmetric matrices. Especially they give higher-order Hadamard variation formulae.
Reviewer: Grozio Stanilov (Sofia)
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 60B20 | Random matrices (probabilistic aspects) |
| 60E05 | Probability distributions: general theory |
| 60E15 | Inequalities; stochastic orderings |
| 62L20 | Stochastic approximation |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Keywords:
Wigner Hermitian matrix; frequent events; Wigner semi-circular law; Dyson sine kernel; Johansson matrices; Lindenberg strategy; four moment theorem; minimax characterization of the eigenvalues; Hadamard first variation formula; empirical spectral distribution; Stieltjes transform; local eigenvalue statistics; random matricesReferences:
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