A Semicircle Law for Derivatives of Random Polynomials

Local semicircle law and complete delocalization for Wigner random matrices

We consider N ×N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N . Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energ...

Semicircle Law for Hadamard Products

In this paper, assuming p/n → 0 as n → ∞, we will prove the weak and strong convergence to the semicircle law of the empirical spectral distribution of the Hadamard product of a normalized sample covariance matrix and a sparsing matrix, which is of the form Ap = 1 √ np (Xm,nX ∗ m,n − σnIm) ◦ Dm, where the matrices Xm,n and Dm are independent and the entries of Xm,n (m × n) are independent, the ...

The Semicircle Law, Free Random Variables and Entropy

Polynomials Orthogonal on the Semicircle

In the above two papers Walter Gautschi, jointly with Henry J. Landau and Gradimir V. Milovanović, investigate polynomials that are orthogonal with respect to a non-Hermitian inner product defined on the upper half of the unit circle in the complex plane. For special choices of the weight function, these polynomials are related to Jacobi polynomials. Their recurrence relation and properties of ...

The local semicircle law for a general class of random matrices

We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij |. As a consequence, we prove t...