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
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parafermionic observable to rule out the first alternative when n ∈ [1, 2] and x = xc(n).