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

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. We show that the pair correlation function is continuous in $\beta$, and provide estimates for its asymptotic decay. We recover the classical explicit formula for the pair correlation function in the $\beta=2$ and $4$ cases. For $\beta=2n$, we derive the power series expansion of the pair correlation function, and express it in terms of a size $n$ linear ordinary differential equation system. We obtain our results by studying the density of the $\operatorname{HP}_{\beta,\delta}$ process, the point process limit of the circular Jacobi beta-ensembles.