## 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

i.e. λ_{1} asymptotically has a normal distribution with expectation (*n*−1)μ+*v*+(σ^{2}/μ) and variance 2σ^{2} (bounded variance!).

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### Cite this article

Füredi, Z., Komlós, J. The eigenvalues of random symmetric matrices.
*Combinatorica* **1, **233–241 (1981). https://doi.org/10.1007/BF02579329

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### AMS subject classification (1980)

- 15 A 52
- 15 A 18
- 05 C 50