Benjamin Eichinger: Universality beyond convergence

In this talk, I will survey recent advances in the study of universality limits of orthogonal polynomials. I will discuss cases where the Christoffel-Darboux kernel admits a power-law scaling limit. Such universality limits typically arise in the bulk or at the edge of the spectrum. However, we show that at accumulation points of spectral gaps, the scaling behavior can be quite different. In particular, we discuss scaling limits where there is not a unique limiting kernel, but rather a full limit cycle. Balanced measures on real Julia sets of arbitrary expanding polynomials provide natural examples of this type of scaling behavior.

This talk is based on joint works with Milivoje Lukić, Brian Simanek, Harald Woracek and Peter Yuditskii.