Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum

Abstract

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 $\frac{1}{n}{K}_n\left(x,x\right)$ 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 ergodic Jacobi matrices with absolutely continuous spectrum and prove that the limit of $\frac{1}{n}{K}_n\left(x,x\right)$ is ${\rho }_{\infty }\left(x\right)∕w\left(x\right)$, where ${\rho }_{\infty }$ is the density of zeros and $w$ is the absolutely continuous weight of the spectral measure.