Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues
1976; Elsevier BV; Volume: 13; Issue: 3 Linguagem: Inglês
10.1016/0024-3795(76)90095-1
ISSN1873-1856
Autores Tópico(s)Advanced Algebra and Geometry
ResumoIt is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let An=(aij), 1⩽i, ⩽n, be the nth section of an infinite Hermitian matrix, {λ(n)}1⩽k⩽n its eigenvalues, and {uk(n)}1⩽k⩽n the corresponding (orthonormalized column) eigenvectors. Let v∗n=(an1,an2,⋯,an,n−1), put Xn(t)=[n(n-1)]-12∑k=1[(n-1)t]|vn∗uf(n-1)|2,0⩽t⩽1 (bookeeping function for the length of the projections of the new row v∗n of An onto the eigenvectors of the preceding matrix An−1), and let finally Fn(x)=n-1(number of λk(n)⩽xn,1⩽k⩽n) (empirical distribution function of the eigenvalues of Ann. Suppose (i) limnannn=0, (ii) limnXn(t)=Ct(0<C<∞,0⩽t⩽1). Then Fn⇒W(·,C)(n→∞),where W is absolutely continuous with (semicircle) densityw(x,C)=(2Cπ)-1(4C-x212for|x|⩽2C0for|x|⩽2C
Referência(s)