Bulk Universality and Clock Spacing of Zeros for Ergodic Jacobi Matrices with A.C. Spectrum

TL;DR

This paper establishes universality and clock spacing of zeros for orthogonal polynomials on the real line in the absolutely continuous spectrum region, linking these properties to kernel convergence and ergodic Jacobi matrices.

Contribution

It proves that universality and clock behavior follow from kernel convergence and boundedness conditions, which are shown to hold for ergodic Jacobi matrices with absolutely continuous spectrum.

Findings

Universality and clock spacing are valid in the a.c. spectral region.

The limit of the scaled kernel is the ratio of zero density to spectral weight.

These properties hold for ergodic Jacobi matrices with a.c. spectrum.

Abstract

By combining some ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for OPRL in the a.c. spectral region is implied by convergence of $\frac{1}{n} K_n(x,x)$ 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 ergodic Jacobi matrices with a.c. spectrum and prove that the limit of $\frac{1}{n} K_n(x,x)$ is $\rho_\infty(x)/w(x)$ where $\rho_\infty$ is the density of zeros and $w$ is the a.c. weight of the spectral measure.