Long-Range Correlation of the Sine$_\beta$ point Process

TL;DR

This paper investigates the decay of correlations in the Sine$_eta$ point process, revealing polynomial decay with an exponent related to $1/\beta$, and extends previous results to all positive $\beta$ and all $k$.

Contribution

It provides a general analysis of correlation decay in Sine$_\beta$ processes for all $\beta > 0$ and all $k$, advancing understanding of their asymptotic behavior.

Findings

Correlation functions decay polynomially with separation

Decay exponent scales as $1/\beta$ for large $\beta$

Results hold for all $\beta > 0$ and $k \geq 1$

Abstract

We study the correlations of the celebrated Sine$_\beta$ point process. This point process arises as the bulk scaling limit of $\beta$-ensembles and has a geometric description through the Brownian carousel, as shown by Valk\'o and Vir\'ag (2009). We establish that the averaged $k$-point truncated correlation functions decay polynomially in the limit of large separation. We show that the decay exponent is of order $1/\beta$ for large $\beta$. This is a step towards a conjecture by Forrester and Haldane regarding the exact asymptotics of the two-point correlation function, a problem recently addressed by Qu and Valk\'o (2025). Our proofs, which rely on a careful analysis of the coupling of diffusions associated with the Brownian carousel, hold for all $\beta >0$ and $k \geq 1$, significantly extending previous results limited to specific values of $\beta$ or $k$.