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Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
13
Issue
3
Identifiers
DOI: 10.1016/0024-3795(76)90095-1

Abstract

Abstract It 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 A n =( a ij ), 1⩽ i, ⩽ n, be the nth section of an infinite Hermitian matrix, { λ ( n) } 1⩽ k⩽ n its eigenvalues, and { u k ( n) } 1⩽ k⩽ n the corresponding (orthonormalized column) eigenvectors. Let v ∗ n=(a n1,a n2,⋯,a n,n−1) , put X n(t)=[n(n-1)] -1 2 ∑ k=1 [(n-1)t] |v n ∗u f (n-1)| 2, 0⩽t⩽1 (bookeeping function for the length of the projections of the new row v ∗ n of A n onto the eigenvectors of the preceding matrix A n−1 ), and let finally F n(x)=n -1( number of λ k (n)⩽x n ,1⩽k⩽n) (empirical distribution function of the eigenvalues of A n n . Suppose (i) lim na nn n =0 , (ii) lim n X n ( t)= Ct(0< C<∞,0⩽ t⩽1). Then F n⇒W(·,C) (n→∞) ,where W is absolutely continuous with (semicircle) density w(x,C)= (2Cπ) -1(4C-x 2 1 2 for |x|⩽2 C 0 for |x|⩽2 C

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