Extend the NILP arctic-curve equation to the full parameter range

Establish that the parametric arctic-curve equation given in Theorem 3.1 for non-intersecting lattice paths remains valid throughout the entire parameter range, rather than only on the parameter intervals corresponding to the southeast and southwest branches.

Background

The paper derives an arctic curve for non-intersecting lattice paths with an arbitrary endpoint distribution and specifies separate parameter ranges for the southeast and southwest branches. Because the arctic curve is analytic in the free-fermion non-intersecting-path model, the authors suggest that the formula may analytically continue beyond those branch-specific ranges. They explicitly identify this extension as a conjecture, and the subsequent touching-path results use the corresponding branch formulas rather than establishing the conjectured global validity.

References

This allowed us to conjecture that the arctic curve equation of Theorem \ref{NILPthm} is still valid over the whole range $t\in $.

— Limit shapes for Domain-Wall (colored) vertex models  (2509.25442 - Francesco et al., 29 Sep 2025) in Section 3.1, immediately following Theorem \ref{NILPthm}