Scaling Limits of Jacobi Matrices and the Christoffel–Darboux Kernel

Abstract

We study scaling limits of deterministic Jacobi matrices, centered around a fixed point $$x_0$$ , and their connection to the scaling limits of the Christoffel–Darboux kernel at that point. We show that in the case when the orthogonal polynomials are bounded at $$x_0$$ , a subsequential limit always exists and can be expressed as a canonical system. We further show that under weak conditions on the associated measure, bulk universality of the CD kernel is equivalent to the existence of a limit of a particular explicit form.