# Universality for conditional measures of the sine point process

## TL;DR

This paper proves a universality result for the sine process, showing that the local correlation kernel converges to the sine kernel as the interval length increases, which advances understanding of the process's universal properties.

## Contribution

The paper establishes that the correlation kernel of the orthogonal polynomial ensemble associated with the sine process converges to the sine kernel, confirming a conjecture and extending universality results.

## Key findings

- Correlation kernel tends to the sine kernel as interval size increases
- Confirms universality of local statistics for the sine process
- Answers a question posed by A.I. Bufetov

## Abstract

The sine process is a rigid point process on the real line, which means that for almost all configurations $X$, the number of points in an interval $I = [-R,R]$ is determined by the points of $X$ outside of $I$. In addition, the points in $I$ are an orthogonal polynomial ensemble on $I$ with a weight function that is determined by the points in $X \setminus I$. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length $|I|=2R$ tends to infinity, thereby answering a question posed by A.I. Bufetov.