Macroscopic loops in the loop \(O(n)\) model at Nienhuis' critical point (Q2659436)

In this paper, the authors consider some of the physicists' prediction for the loop $O(n)$ model and study them in a wide regime of parameter. In particular, they prove rigorously that the loop $O(n)$ model, for $n$ in $[1,2]$, on the hexagonal lattice exhibits macroscopic loops at criticality (for critical edge-weight, so-called dilute critical regime), using a new FKG property and gluing techniques inspired by recent progress made while investigating scaling properties in random-cluster and loop models.