Wigner Semicircle Research Articles

Number of energy levels outside Wigner's semicircle

The total probability outside Wigner's (1962) semicircle is calculated using Mehler's formula for the Hermite polynomials. In the limit N to infinity , N being the dimension of the random Gaussian Hermitian matrix it is shown that this probability is 0.05N-32/. Some implications of this small probability are discussed.

On the expansion of the single eigenvalue probability density function

The probability density function of the single eigenvalue is expanded in terms of the reciprocal of the dimension of the matrix using Bessel functons. It is shown that for the new matrix ensembles this expansion gives Wigner's semicircle centered at the mean value of the matrix elements plus terms of the order ofN−1, whereN is the dimension of the matrix.

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ 1, …, λ n the eigenvalue-distribution is defined by G(x, A) := 1 n · number {λ i : λ i ⩽ x} for all real x. Let A n for n = 1, 2, … be an n × n matrix, whose entries a ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, R , p) with the same distribution F a . Suppose that all moments E | a | k, k = 1, 2, … are finite, E a=0 and E | a | 2. Let M(A)= ∑ σ=1 s θ σP σ(A,A ∗) with complex numbers θ σ and finite products P σ of factors A and A ∗ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M( A). The following limit theorem is proved: There exists a distribution function G 0( x) = G 1 x) + G 2( x), where G 1 is a step function and G 2 is absolutely continuous, such that with probability 1 G(x, M( A n n 1 2 )) converges to G 0( x) as n → ∞ for all continuity points x of G 0. The density g of G 2 vanishes outside a finite interval. There are only finitely many jumps of G 1. Both, G 1 and G 2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M( A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices ( E. P. Wigner, Random matrices in physics, SIAM Review 9 (1967) , 1–23). The examples A rA ∗r , A r + A ∗r , A rA ∗r ± A ∗rA r , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form lim sup n→∞ ∑ i=1 n |γ i (n)| 2⧹‖A n‖ 2⩽ 0.8228… of Schur's inequality for the eigenvalues λ i ( n) of A n holds. Consequently random matrices do not tend to be normal matrices for large n.

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

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

Two-body random hamiltonian and level density

It is shown numerically that strong deviations from Wigner's semi-circle law for the level probability density of a random matrix are obtained if the two-body nature of the hamiltonian is taken into account. Instead of Wigner's law the probability density approaches closely a normal (gaussian) distribution.

On the asymptotic distribution of the eigenvalues of random matrices

Abstract : A generalization of E. P. Wigner's semi-circle law concerning the distribution of the eigenvalues of large symmetric matrices A = (a sub ij) whose entries a sub ij are for i > or = j independent and identically distributed random variables. (Author)