Abstract
We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl $m$-function at the point. We show that bulk universality of the Christoffel–Darboux kernel holds for any point where the imaginary part of the $m$-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel–Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding $m$-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with $2\times 2$ transfer matrices such as continuum Schrödinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.
