Combinatorica

, Volume 1, Issue 3, pp 233–241 | Cite as

The eigenvalues of random symmetric matrices

  • Z. Füredi
  • J. Komlós
Article

Abstract

LetA=(aij) be ann ×n matrix whose entries forij are independent random variables andaji=aij. Suppose that everyaij is bounded and for everyi>j we haveEaij=μ,D2aij2 andEaii=v.

E. P. Wigner determined the asymptotic behavior of the eigenvalues ofA (semi-circle law). In particular, for anyc>2σ with probability 1-o(1) all eigenvalues except for at mosto(n) lie in the intervalI=(−cn,cn).

We show that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalue λ1 is outsideI, and
$$\lambda _1 = \frac{{\Sigma _{i,j} a_{ij} }}{n} + \frac{{\sigma ^2 }}{\mu } + O\left( {\frac{I}{{\sqrt n }}} \right)$$
i.e. λ1 asymptotically has a normal distribution with expectation (n−1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).

AMS subject classification (1980)

15 A 52 15 A 18 05 C 50 

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Copyright information

© Akadémiai Kiadó 1981

Authors and Affiliations

  • Z. Füredi
    • 1
  • J. Komlós
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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