DMCA
The eigenvalues of random symmetric matrices (1981)
| Venue: | Combinatorica |
| Citations: | 179 - 0 self |
BibTeX
@ARTICLE{Füredi81theeigenvalues,
author = {Z. Füredi and J. Komlós},
title = {The eigenvalues of random symmetric matrices},
journal = {Combinatorica},
year = {1981},
volume = {1},
pages = {233--241}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
Let A:(at;) be an zxn matrix whose entries for i=j are independent random variables and ai;:aii. Suppose that every a;; is bounded and for eveÍy i>j we have Eaii:!, D2ai,:62 and Eaii-v. E. P. Wigner determined the asymptotic behavior of the eigenvalues of I (semi-circle law). In particular, for any c>2a with probability I- o (1) all eigenvalues except for at most o (n) lie in the interval t:(_,Íi ' c|i). We show that with probability 1-o(1) a/ / eigenvalues belong to the above interval 1if p:0, while in case p=0 only the largest eigenvalue,lt is outside /, and n,: , Zl.iaii- ' "'-o í-L],,i "ll;) i.e. l, asymptotically has a normal distribution with expectation (n-1)p*v*(o2lp) and, variance 2a'z (bounded variance!). 1. fntuoduction E. P. Wigner published in 1955 his famous semi-circle law for the distribution of eigenvalues of random sFnmetric matrices (used in quantum mechanics). Here we recall the following generalization due to L. Arnold [1] (see also U. Grenander [3]): Let A:(ai), l=i, j<n, be an nXn symmetric matrix where the entries arj, i=j, are independent real-valued random variables. Furthermore, the a, with i>.i are required to have the same distribution function 4 while all the ú7'í possess the same distribution G. Let)r>),r>...4A, be the eigenvalues of A (all real by the symmetry of l). We also use the notation l;('4). Further, WA'n or simply W, denoÍes their empirical distribution function, i.e. Wa,,(x):: (number of eigenvalues =x)ln' One is interested in the limiting behavior of the sequence W,, of random distribution functions as n*@.
Keyphrases
random symmetric matrix nxn symmetric matrix normal distribution independent real-valued random variable random distribution function bounded variance famous semi-circle law empirical distribution function semi-circle law quantum mechanic limiting behavior random sfnmetric matrix asymptotic behavior independent random variable distribution function zxn matrix following generalization
