On the pair correlation function of the Sine_β process

We study the Sine_β process, the bulk point process scaling limit of beta-ensembles. We provide a representation of its pair correlation function for all $β>0$ via a stochastic differential equation. We show that the pair correlation function is continuous in $β$, and provide estimates for its asymptotic decay. We recover the classical explicit formula for the pair correlation function in the $β=2$ and $4$ cases. For $β=2n$, 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 HPβ,δ process, the point process limit of the circular Jacobi beta-ensembles.