# Large Deviations and the Lukic Conjecture

## TL;DR

This paper employs large deviation techniques to establish higher order sum rules for orthogonal polynomials on the unit circle, providing partial proof of Lukic's conjecture in a specific singular case, which contrasts with the failure of Simon’s conjecture.

## Contribution

It proves a significant part of Lukic's conjecture for the case of two singular points, advancing understanding of sum rules and orthogonal polynomials.

## Key findings

- Proves one half of Lukic's conjecture for two singular points
- Supports the validity of Lukic's conjecture where Simon's fails
- Uses large deviation approach to sum rules in orthogonal polynomial theory

## Abstract

We use the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault to prove higher order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two singular points, one simple and one double. This is important because it is known that the conjecture of Simon fails in exactly this case, so this paper provides support for the idea that Lukic's replacement for Simon's conjecture might be true.