On the pair correlation function of the Sine_β process

2025/09/18 by Yan Qu, Qu, Yahui, Benedek Valkó +1
Decision Sciences · Economics, Econometrics and Finance · #Complex Systems and Time Series Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2509.15446

openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 HP_β,δ process, the point process limit of the circular Jacobi beta-ensembles.

Citations

• DLR equations and rigidity for the Sine-beta process
• CLT for fluctuations of linear statistics in the Sine-beta process
• The maximum deviation of the Sine_β counting process
• Hard edge tail asymptotics
• On n-point correlations in the log-gas at rational temperature

Related

• The pair correlation function of the Sine₆ process
• Operator level limit of the circular Jacobi β-ensemble
• Point Processes and the Infinite Symmetric Group. Part II: Higher Correlation Functions
• Asymptotic correlations with corrections for the circular Jacobi β-ensemble
• The maximum deviation of the Sine_β counting process