Nienhuis critical-point conjecture for the hexagonal-lattice loop O(n) model

Prove that the critical point of the loop O(n) model on the hexagonal lattice occurs at x_c = 1/sqrt(2 + sqrt(2-n)) for n in [-2,2], as predicted by Nienhuis.

Background

The paper situates the fully packed loop O(n) model on planar triangulations within the broader theory of loop models and their critical behavior. It recalls Nienhuis’s prediction that, for the loop O(n) model on the hexagonal lattice, the critical point coincides with the self-dual point and has an explicit coupling constant.

The authors note that this prediction remains unresolved and use it as context for the analogous self-dual models on planar maps studied in the paper. This is an external open problem rather than a question answered by the paper.

References

For the related loop $O(n)$ model on the hexagonal lattice, the equality between critical and self-dual point led Nienhuis to predict () that the critical point occurs at $x = x_c = 1/ \sqrt{2 + \sqrt{2-n}$ for $n \in (0,2]$, and in fact $n \in [-2, 2]$; this remains a famous open problem in this field (see for a thorough survey).

— Critical behaviour of the fully packed loop-$O(n)$ model on planar triangulations  (2512.05867 - Berestycki et al., 5 Dec 2025) in Section 1, Introduction