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 x(c)(n) = 1/root 2 + root 2 - n. For 0 < n <= 2, the transition line has been further conjectured to separate a regime with short loops when x < x(c)(n) from a regime with macroscopic loops when x >= x(c)(n).In this paper, we prove that for n is an element of[1, 2] and x = x(c)(n), the loop O(n) model exhibits macroscopic 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/root n. This property implies, 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 is an element of[1, 2] and x = x(c)(n).