On the pair correlation function of the Sine${}_{\hat{I}}^2$ process

Abstract

We study the Sine${}_{\hat{I}}^2$ process, the bulk point process scaling limit of beta-ensembles. We provide a representation of its pair correlation function for all ${\hat{I}}^2>0$ via a stochastic differential equation. We show that the pair correlation function is continuous in ${\hat{I}}^2$, and provide estimates for its asymptotic decay. We recover the classical explicit formula for the pair correlation function in the ${\hat{I}}^2=2$ and $4$ cases. For ${\hat{I}}^2=2n$, 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 ${\mathrm{HP}}_{{\hat{I}}^2,\hat{I}´}$ process, the point process limit of the circular Jacobi beta-ensembles.

32 pages