Georgia Institute of Technology College of Sciences

The local spacing of zeros of orthogonal polynomials is studied using scaling limits of Christoffel--Darboux kernels. Different limit kernels are associated with different universality classes, e.g. sine kernel with bulk universality and locally asymptotically uniform zero spacing. In recent years, new results have been obtained by using the de Branges theory of canonical systems. This includes necessary and sufficient conditions for a family of scaling limits corresponding to homogeneous de Branges spaces; this family includes bulk universality, hard edge universality, jump discontinuities in the weight, and other notable universality classes. It also includes local behaviors beyond scaling limits. The talk is based on joint works with Benjamin Eichinger, Brian Simanek, Harald Woracek, Peter Yuditskii.