# Scaling Limits of Jacobi Matrices and the Christoffel-Darboux Kernel

## TL;DR

This paper investigates the scaling limits of Jacobi matrices and the Christoffel-Darboux kernel, establishing conditions for the existence of limits and their relation to universality in orthogonal polynomial systems.

## Contribution

It demonstrates that subsequential limits of scaled Jacobi matrices can be expressed as canonical systems and links kernel universality to explicit limit forms.

## Key findings

- Subsequential limits exist when polynomials are bounded at a point.
- Limits can be represented as canonical systems.
- Bulk universality is equivalent to the existence of specific kernel limits.

## Abstract

We study scaling limits of deterministic Jacobi matrices at 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 that 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.