Stability of Asymptotics of Christoffel-Darboux Kernels

We study the stability of convergence of the Christoffel-Darboux kernel, associated with a compactly supported measure, to the sine kernel, under perturbations of the Jacobi coefficients of the measure. We prove stability under variations of the boundary conditions and stability in a weak sense under ℓ¹ and random ℓ² diagonal perturbations. We also show that convergence to the sine kernel at x implies that μ({x}) = 0.