Complete large-distance expansion of the Sine-beta two-point correlation

Establish a complete large-distance asymptotic expansion of the two-point correlation function of the Sine-beta point process for arbitrary beta>0, beyond the leading asymptotics proved in the paper.

Background

The paper proves the leading large-distance asymptotics of the truncated two-point correlation function of the Sine-beta point process for every beta>0, including the BKT transition at beta=2 and the exact oscillatory coefficient in the beta>2 regime. A complete asymptotic expansion would describe all subsequent algebraic and oscillatory terms for general beta.

The authors note that substantially more is already known for even integer beta through Forrester’s work, but the corresponding complete expansion for arbitrary positive beta remains unresolved.

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