Operator limit of the circular beta ensemble

TL;DR

This paper establishes a detailed coupling between finite circular beta ensembles and their infinite limit, providing bounds on operator and process distances, and analyzing beta-dependence in the sine process.

Contribution

It introduces explicit operator couplings for circular beta ensembles and their limits, with bounds and beta-dependence estimates, advancing understanding of their asymptotic behavior.

Findings

Explicit bounds on operator and point process distances.

Coupling construction for finite and infinite ensembles.

Beta-dependence estimates in the sine process.

Abstract

We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate on the beta-dependence in the $\text{Sine}_{\beta}$ process.