### Abstract

Let A=(a_{ ij} ) be an n ×n matrix whose entries for i≧j are independent random variables and a_{ ji} =a_{ ij} . Suppose that every a_{ ij} is bounded and for every i>j we have Ea_{ ij} =μ, D^{ 2} a_{ ij} =σ^{2} and Ea_{ ii} =v. E. P. Wigner determined the asymptotic behavior of the eigenvalues of A (semi-circle law). In particular, for any c>2σ with probability 1-o(1) all eigenvalues except for at most o(n) lie in the interval I=(-c√n, c√n). We show that with probability 1-o(1)all eigenvalues belong to the above interval I if μ=0, while in case μ>0 only the largest eigenvalue λ_{1} is outside I, and {Mathematical expression} i.e. λ_{1} asymptotically has a normal distribution with expectation (n-1)μ+v+(σ^{2}/μ) and variance 2σ^{2} (bounded variance!).

Original language | English |
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Pages (from-to) | 233-241 |

Number of pages | 9 |

Journal | Combinatorica |

Volume | 1 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 1981 |

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### Keywords

- AMS subject classification (1980): 15A52, 15A18, 05C50

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*1*(3), 233-241. https://doi.org/10.1007/BF02579329