  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 $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a
critical parameter $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}$. For $0<n\le 2$, 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 $x\ge x_c(n)$.
  In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$
model exhibits macroscopic loops. This is the first instance in which a loop
$O(n)$ model with $n\neq 1$ is shown to exhibit such behaviour. A main tool in
the proof is a new positive association (FKG) property shown to hold when $n
\ge 1$ and $0<x\le\frac{1}{\sqrt{n}}$. 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 $x=x_c(n)$ and $n\in[1,2]$.
