The eigenvalues of random symmetric matrices

Z. Füredi, J. Komlós

Research output: Contribution to journalArticle

324 Citations (Scopus)

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 ij2 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 λ1 is outside I, and {Mathematical expression} i.e. λ1 asymptotically has a normal distribution with expectation (n-1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).

Original languageEnglish
Pages (from-to)233-241
Number of pages9
JournalCombinatorica
Volume1
Issue number3
DOIs
Publication statusPublished - Sep 1 1981

    Fingerprint

Keywords

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

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Cite this