Bulk Universality and Clock Spacing of Zeros 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.

References in corpus (3)

• Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures
• A New Approach to Universality at the Edge of the Spectrum
• Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operators

Cited by in corpus (9)

• An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles
• Stability of Asymptotics of Christoffel-Darboux Kernels
• The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles
• Poisson statistics for 1d Schrödinger operators with random decaying potentials
• The Nevai Condition
• Finite Gap Jacobi Matrices, I. The Isospectral Torus
• Sine kernel asymptotics for a class of singular measures
• Level repulsion for Schroedinger operators with singular continuous spectrum
• Clock statistics for 1d Schrödinger operators