Rigidity of the $\operatorname{Sine}_{\beta}$ process

TL;DR

This paper proves that the Sine_beta process, a limit of the Circular Beta Ensemble, exhibits rigidity, meaning the number of points in a set is determined by points outside it.

Contribution

It establishes the rigidity property for the Sine_beta process, extending the understanding of point process behaviors in random matrix theory.

Findings

Proves rigidity of the Sine_beta process.

Shows the number of points in a set is almost surely determined by outside points.

Extends rigidity results to a broader class of processes.

Abstract

We show that the $\operatorname{Sine}_{\beta}$ point process, defined as the scaling limit of the Circular Beta Ensemble when the dimension goes to infinity, and generalizing the determinantal sine-kernel process, is rigid in the sense of Ghosh and Peres: the number of points in a given bounded Borel set $B$ is almost surely equal to a measurable function of the position of the points outside $B$.