Delocalisation and Continuity in 2D: Loop  , Six-Vertex, and Random-Cluster Models

We prove the existence of macroscopic loops in the loop  $$ model with $$ 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 $$ is the critical point. We also prove delocalisation in the six-vertex model with $$ . This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for $$ relying neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory. Our approach goes through a novel FKG property required for the non-coexistence theorem of Zhang and Sheffield, which is used to prove delocalisation all the way up to the critical point. We also use the $$ -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 $$ and $$ . This is consistent with the conjecture that the scaling limit is the Gaussian free field.