Stability of Asymptotics of Christoffel-darboux Kernels

We study the stability of convergence of the ChristoffelDarboux 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 ` and random ` diagonal perturbations. We also show that convergence to the sine kernel at x implies that μ({x}) = 0.