Precise pointwise asymptotics for higher correlation functions

Determine precise pointwise large-distance asymptotics for higher-order correlation functions of the Sine-beta point process for arbitrary beta>0.

Background

The paper focuses on the two-point correlation function and establishes its leading large-distance behavior for all beta>0. Although spatially averaged decay results for correlations of every order are cited, precise pointwise asymptotics for higher-order correlation functions are not obtained.

The unresolved problem concerns extending the type of pointwise asymptotic information proved here for the two-point function to higher correlation functions of the Sine-beta point process.

References

Obtaining a complete large-distance expansion of the second correlation function for arbitrary \beta>0 (of course for even integer \beta this and much more has already been proven in the works of Forrester), or precise pointwise asymptotics for higher correlation functions, remains open and very interesting.

— BKT transition for $\mathsf{Sine}_β$  (2610.13484 - Assiotis, 9 Oct 2026) in Introduction, immediately following Theorem 1