  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
$\ell^1$ and random $\ell^2$ diagonal perturbations. We also show that
convergence to the sine kernel at $x$ implies that $\mu(\{x\})=0$.
