Abstract
We prove the existence of macroscopic loops in the loop O(2) model with ½ ≤ x2 ≤ 1 or, equivalently, delocalisation of the associated integer-valued Lipschitz function on the triangular lattice. This settles one side of the conjecture of Fan, Domany, and Nienhuis (1970 s-1980 s) that x2 = ½ is the critical point. We also prove delocalisation in the six-vertex model with 0 < a, b ≤ c ≤ a + b . This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for 1 ≤ q ≤ 4 relying neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory. Our approach goes through a novel FKG property required for the non-coexistence theorem of Zhang and Sheffield, which is used to prove delocalisation all the way up to the critical point. We also use the T -circuit argument in the case of the six-vertex model. Finally, we extend an existing renormalisation inequality in order to quantify the delocalisation as being logarithmic, in the regimes ½ ≤ x2 ≤ 1 and a = b ≤ c ≤ a + b . This is consistent with the conjecture that the scaling limit is the Gaussian free field.