Dense-phase non-simplicity and mutual touching of loop O(n) interfaces

Characterize the scaling limit of loops in the dense phase of the fully packed loop O(n) model on planar triangulations and prove that the loops are non-simple and touch one another in that limit.

Background

The paper proves that the partition function has asymptotics characteristic of the non-generic critical, dense phase of the loop O(n) model. In this phase, the authors state a conjectural geometric description of the scaling limit: loops should fail to be simple and should touch each other.

The paper establishes the critical exponent but does not establish this geometric scaling-limit behavior. The explicit uncertainty marker identifies the claim as an unresolved conjecture.

References

More precisely, it lies in the so-called dense phase of the model, where loops are conjectured to be non-simple and to touch each other in the scaling limit.

— 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, immediately after Corollary 1