On the pair correlation function of the Sine${}_{\beta }$ process

Abstract

We study the Sine${}_{\beta }$ process, the bulk point process scaling limit of beta-ensembles. We provide a representation of its pair correlation function for all $\beta >0$ via a stochastic differential equation, and also via an infinite system of differential equations. We show that the pair correlation function is continuous in $\beta$, and provide estimates for its asymptotic decay. When $\beta =2n$ is an even integer, we characterize the pair correlation function through a linear system of $n$ ordinary differential equations and derive the function's power series expansion. Our results recover the classical explicit formula for the pair correlation function in the $\beta =2$ and $4$ cases. We obtain our results by studying the one-point function of the ${\mathrm{HP}}_{\beta ,\delta }$ process, the point process limit of the circular Jacobi beta-ensemble.

arXiv comment46 pages, revised version based on referee feedback. Theorem 2 is new, the statement of Theorem 4 has been strengthened and the proof has been revised. Updated the introduction with an overview of recent results, reorganized sections to improve presentation, and added an Appendix