Delocalisation and Continuity in 2D: Loop \(\textrm{O}(2)\), Six-Vertex, and Random-Cluster Models

Abstract

We prove the existence of macroscopic loops in the loop \(\textrm{O}(2)\) 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 (1970 s–1980 s) 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.