Delocalisation and continuity in 2D: loop O(2), six-vertex, and random-cluster models

We prove the existence of macroscopic loops in the loop O(2) model with $\frac{1}{2}\le x^2\le 1$ 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 (1970s-80s) that $x^2=\frac{1}{2}$ is the critical point. We also prove delocalisation in the six-vertex model with $0<a,b\le c\le a+b$. This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for $1\le q\le 4$ 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 $\mathbb{T}$-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 $\frac{1}{2}\le x^2\le 1$ and $a=b\le c\le a+b$. This is consistent with the conjecture that the scaling limit is the Gaussian free field.