Ergodic Jacobi Matrices and Conformal Maps

Ergodic Jacobi Matrices and Conformal Maps

We study structural properties of the Lyapunov exponent γ and the density of states k for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function w = −γ + iπk as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.

Individual Ergodic Theorem for Unitary Maps of Random Matrices

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices. 1. Notation and main result 1.1. Let (Ω,F , μ) be a probability space, and let H denote the space of all d × d matrices with entries in the complex space L2(Ω, μ). H is a Hilbert space with the inner product (1) 〈AB〉 = Φ(AB∗), where φ(A) = ∫ Ω tr A(ω)μ(dμ), and tr denotes the normalized trace o...

Two modified Jacobi methods for M-matrices

Jacobi matrices

We discuss the relationships among Jacobi matrices, orthogonal polynomials, spectral measure, moments, minors, Gaussian quadrature, resolvents and continued fractions in the simplest setting, namely the finite-dimensional one. The formal structure is essentially the same as that in the infinite-dimensional setting, where it leads into the rich analytic world of orthogonal polynomials on the rea...

Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with Absolutely Continuous Spectrum

By combining ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for orthogonal polynomials on the real line in the absolutely continuous spectral region is implied by convergence of 1 n Kn.x; x/ for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for erg...