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

The loop O(n) model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin O(n) model. It has been predicted by Nienhuis that for 0 ≤ n ≤ 2, the loop O(n) model exhibits a phase transition at a critical parameter xc(n) = 1/√2 + √2 − n. For 0 < n ≤ 2, the transition line has been further conjectured to separate a regime with short loops when x < xc(n) from a regime with macroscopic loops when x ≥ xc(n). In this paper, we prove that for n ∈ [1, 2] and x = xc(n), the loop O(n) model exhibits macro- scopic loops. Apart from the case n = 1, this constitutes the first regime of parameters for which macroscopic loops have been rigorously established. A main tool in the proof is a new positive association (FKG) property shown to hold when n ≥ 1 and 0 < x ≤ 1/√n. This property im- plies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box- crossing property). We develop a “domain gluing” technique which allows us to employ Smirnov’s parafermionic observable to rule out the first alternative when n ∈ [1, 2] and x = xc(n).