In this talk, I will present joint work with Benjamin
Eichinger and Brian Simanek: a new 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 2x2 transfer matrices such as continuum
Schrodinger and Dirac operators. We also obtain analogous results for
orthogonal polynomials on the unit circle.