Uniform edge asymptotics at the arctic curves

Develop a uniform edge analysis of the outer frozen–liquid and inner liquid–gas arctic curves by constructing the required local edge parametrices and determining whether the liquid–gas edge exhibits band- and matrix-dependent structure beyond the generic Airy scaling associated with coalescing saddles.

Background

The paper derives the outer frozen–liquid and inner liquid–gas boundaries through a bulk saddle-point analysis. That analysis is explicitly nonuniform near an arctic boundary, where two saddles coalesce and the bulk correlation-kernel formulas no longer apply directly.

The authors identify the Chester–Friedman–Ursell reduction as the generic mechanism suggesting Airy scaling, but leave unresolved the construction of a uniform treatment for both boundaries and the possible additional band and matrix structure at the liquid–gas interface. This problem concerns the precise edge kernels and scaling limits near the two arctic curves.

References

Several more immediate questions remain open. First, the analysis here is a bulk analysis. A uniform treatment of the outer and inner arctic curves requires local edge parametrices. A generic coalescence of two saddles is described by the Chester--Friedman--Ursell reduction and suggests Airy scaling, as at the ordinary arctic boundary ; the liquid--gas edge may nevertheless retain additional band and matrix structure.

— Arctic Curves and a Gapped Gas Phase in a Two-Band Free-Fermion Chain  (2609.21004 - Jordan et al., 17 Sep 2026) in Section 9, paragraph beginning “Several more immediate questions remain open”

Several more immediate questions remain open. First, the analysis here is a bulk analysis. A uniform treatment of the outer and inner arctic curves requires local edge parametrices. A generic coalescence of two saddles is described by the Chester--Friedman--Ursell reduction and suggests Airy scaling, as at the ordinary arctic boundary ; the liquid--gas edge may nevertheless retain additional band and matrix structure. Second, the closing-gap regime is nonuniform: the gas region shrinks while its correlation length diverges. The double scaling limit $R\Delta2=O(1)$ should describe the crossover between the homogeneous arctic-circle geometry and a resolved gas region.

— Arctic Curves and a Gapped Gas Phase in a Two-Band Free-Fermion Chain  (2609.21004 - Jordan et al., 17 Sep 2026) in Section 9, paragraph beginning “Several more immediate questions remain open”