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 macroscopic loops. This is the first instance in which a loop O(n) model with n = 1 is shown to exhibit such behavior. 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 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 ∈ [1, 2] and x = xc(n).

Abstract. We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z d for d ≥ 2. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x ∈ Z d , this probability decreases faster than n −1/4+ε for any ε > 0. When ||x|| = 1, we thus obtain a bound on the probability that self-avoiding walk is a polygon.

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