Scaling Limits of Jacobi Matrices and the Christoffel–Darboux Kernel

We study scaling limits of deterministic Jacobi matrices, centered around a fixed point x, 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, 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.