# Delocalisation and Continuity in 2D: Loop O(2), Six-Vertex, and Random-Cluster Models

## TL;DR

This paper proves the existence of macroscopic loops in certain 2D statistical physics models, confirming a long-standing conjecture about their critical points.

## Contribution

A novel proof of delocalisation in the loop O(2) and six-vertex models without relying on integrability or Russo–Seymour–Welsh theory.

## Key findings

- Macroscopic loops exist in the loop O(2) model for $\frac{1}{2} \le x^2 \le 1$.
- Delocalisation is proven in the six-vertex model with $0 < a, b \le c \le a + b$.
- A new proof of phase transition continuity in 2D random-cluster and Potts models for $1 \le q \le 4$.

## Abstract

We prove the existence of macroscopic loops in the loop \documentclass[12pt]{minimal}
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\begin{document}$$\textrm{O}(2)$$\end{document}O(2) model with \documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{2}\le x^2\le 1$$\end{document}12≤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 \documentclass[12pt]{minimal}
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\begin{document}$$x^2 = \frac{1}{2}$$\end{document}x2=12 is the critical point. We also prove delocalisation in the six-vertex model with \documentclass[12pt]{minimal}
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\begin{document}$$0<a,\,b\le c\le a+b$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$1\le q\le 4$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\frac{1}{2}\le x^2\le 1$$\end{document}12≤x2≤1 and \documentclass[12pt]{minimal}
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\begin{document}$$a=b\le c\le a+b$$\end{document}a=b≤c≤a+b. This is consistent with the conjecture that the scaling limit is the Gaussian free field.