On the spectrum of random matrices

On the Law of Addition of Random Matrices

“…[9,10,11,12,13,14,20,21,22,25,26]). We list the properties of the Stieltjes transform that we will need below (see e.g.…”

“…The method is based on certain differential identities for expectations of smooth matrix functions with respect to the normalized Haar measure of U (n) ( or O(n) ) and on elementary matrix identities, the resolvent identity first of all. The basic idea is the same as in [17,20]: to study not the moments of the counting measure, as it was proposed in the pioneering paper by Wigner [35], but rather its Stieltjes (called also the Cauchy or the Borel) transform, playing the role of appropriate generating (or characteristic) function of the moments (the measure). However, the technical implementation of the idea in this paper is different and simpler then in [17,20] (see Remark 1 after Theorem 2.1).…”

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

“…In the corresponding QVE (2.1), we have X = [0, 1] with uniform measure, s xy ≡ λ and a x the smooth limiting profile of the diagonal entries of A. The average generating density ρ S,a equals the asymptotic density of the eigenvalues as the dimension of H approaches infinity [33]. In particular, Theorem 2.6 restricts the possible singularities of the limiting eigenvalue density to at most third order.…”

“…For the proof, we use the QVE to obtain 33) for every x, y ∈ X and τ ∈ R. Suppose now that (6.32) is not true, so that the set…”

Operator-valued Semicircular Elements: Solving A Quadratic Matrix Equation with Positivity Constraints

“…Note that for x = 0 we have G 1 (z) = 1/z and (3.2) reduces to the fixed point version of (1.2). In the scalar-valued case A = C, equation (3.2) was derived by Pastur [7], describing a 'deformed semicircle'. The same arguments as before show that for any fixed z with positive imaginary part there exists exactly one solution of (3.2) whose imaginary part is strictly negative.…”