标题： 二维中的离域性和连续性：环O(2)、六顶点和随机簇模型 显示英文标题

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

摘要： 我们证明了在环O(2)模型中存在宏观环，其中$\frac12\leq x^2\leq 1$或等价地，关联的整数值Lipschitz函数在三角形格点上的非定位性。这解决了Fan、Domany和Nienhuis（1970年代-80年代）猜想的一方面，即$x^2 = \frac12$是临界点。我们还证明了六顶点模型中$0<a,\,b\leq c\leq a+b$的非定位性。这提供了一种新的证明方法，证明二维随机簇和Potts模型相变的连续性，对于$1\leq q\leq 4$无需依赖可积性工具（如 parafermionic 观测量、Bethe Ansatz），也不依赖Russo-Seymour-Welsh理论。我们的方法通过了一个用于Zhang和Sheffield非共存定理的新FKG性质，该性质被用来一直证明到临界点的非定位性。我们还在六顶点模型的情况下使用了$\mathbb T$回路论证。最后，我们扩展了一个现有的重正化不等式，以量化非定位性在区域$\frac12\leq x^2\leq 1$和$a=b\leq c\leq a+b$中为对数级别。这与标度极限是高斯自由场的猜想是一致的。 显示英文摘要

摘要： We prove the existence of macroscopic loops in the loop O(2) model with $\frac12\leq x^2\leq 1$ 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 $x^2 = \frac12$ is the critical point. We also prove delocalisation in the six-vertex model with $0<a,\,b\leq c\leq a+b$. This yields a new proof of continuity of the phase transition in the random-cluster and Potts models in two dimensions for $1\leq q\leq 4$ 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 $\frac12\leq x^2\leq 1$ and $a=b\leq c\leq a+b$. This is consistent with the conjecture that the scaling limit is the Gaussian free field.