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The loop $O(n)$ model is a family of probability measures on collections ofnon-intersecting loops on the hexagonal lattice, parameterized by a loop-weight$n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, themodel exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$:when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geqx_c(n)$, the loops are macroscopic.  In this paper, we prove three results regarding the existence of long loopsin the loop $O(n)$ model:  - In the regime $(n,x) \in [1,1+\delta) \times (1- \delta, 1]$ with $\delta>0$ small, a configuration sampled from a translation-invariant Gibbs measurewill either contain an infinite path or have infinitely many loops surroundingevery face. In the subregime $n \in [1,1+\delta)$ and $x \in(1-\delta,1/\sqrt{n}]$ our results further imply Russo--Seymour--Welsh theory.This is the first proof of the existence of macroscopic loops in a positivearea subset of the phase diagram.  - Existence of loops whose diameter is comparable to that of a finite domainwhenever $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of theantiferromagnetic regime of the Ising model on the triangular lattice.  - Existence of non-contractible loops on a torus when $n \in [1,2], x=1$.  The main ingredients of the proof are: (i) the `XOR trick': if $\omega$ is acollection of short loops and $\Gamma$ is a long loop, then the symmetricdifference of $\omega$ and $\Gamma$ necessarily includes a long loop as well;(ii) a reduction of the problem of finding long loops to proving that apercolation process on an auxiliary planar graph, built using theChayes--Machta and Edwards--Sokal geometric expansions, has no infiniteconnected components; and (iii) a recent result on the percolation threshold ofBenjamini--Schramm limits of planar graphs.

Macroscopic loops in the loop O(n) model via the XOR trick