Convergence of loop O(n) interfaces to conformal loop ensembles

Prove convergence of the loop $O(n)$ model at macroscopic or critical parameter values to the appropriate conformal loop ensemble $\mathrm{CLE}_\kappa$ for the full conjectured range of parameters.

Background

The paper describes macroscopic loop behavior as weaker than scale invariance and relates it to a conjectured conformal scaling limit. The parameter κ\kappa depends on (n,x)(n,x), and rigorous convergence is reported only for Bernoulli site percolation and the critical Ising model. Thus, establishing the conjectured CLE convergence for the broader loop O(n)O(n) family remains unresolved.

References

The macroscopic behavior is a weaker version of scale invariance and is in agreement with the conjectured convergence to the conformal loop ensemble (CLE) with an appropriate~$\kappa = \kappa (n,x)$; seeSection~5.6. The convergence has been proved only for Bernoulli site percolation ($n=x=1$) and for the critical Ising model~($n=1,\, x=1/{\sqrt{3}$).

— Planar percolation and the loop O(n) model  (2508.20917 - Glazman et al., 28 Aug 2025) in Section 1, subsection “Loop $O(n)$ model”