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： 我们在环O(2)模型中证明了宏观环的存在，该模型具有$\frac12 \leq x^2 \leq 1$。这表明在三角形格子上的整数值Lipschitz函数的对数非定位性，并解决了Fan、Domany和Nienhuis的猜想的一方面：他们预测在1970年代至1980年代，$x^2 = \frac12$是定位-非定位相变的临界点。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。这为二维随机簇和Potts模型相变的连续性提供了新的证明，对于$1 \leq q \leq 4$。我们既不依赖可积性工具（如参数可观测量、Bethe假设），也不依赖Russo-Seymour-Welsh理论。我们的方法通过满足平面对偶性、联合FKG和区域马尔可夫性质的新图形表示。这使我们能够使用Zhang和Sheffield的非共存定理，并一直证明到临界点的非定位性。在六顶点模型中，我们还使用了$\mathbb T$回路论证。最后，在区域$\frac12 \leq x^2 \leq 1$和$a = b \leq c \leq a + b$中，我们利用额外的对称性扩展现有的重整化不等式，并建立给定点的高度的对数波动。这与这些高度函数的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 $\frac12 \leq x^2 \leq 1$. This implies a logarithmic delocalisation of an integer-valued Lipschitz function on the triangular lattice and settles one side of the conjecture of Fan, Domany, and Nienhuis: they predicted in the 1970s-80s that $x^2 = \frac12$ is the critical point for the localisation-delocalisation transition. 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 $1 \leq q \leq 4$. We rely neither on integrability tools (parafermionic observables, Bethe Ansatz), nor on the Russo-Seymour-Welsh theory. Our approach goes through novel graphical representations that satisfy planar duality, the joint FKG and the domain Markov properties. This enables us to use the non-coexistence theorem of Zhang and Sheffield and prove delocalisation all the way up to the critical point. In the six-vertex model, we additionally use the $\mathbb T$-circuit argument. Finally, in the regimes $\frac12 \leq x^2 \leq 1$ and $a = b \leq c \leq a + b$, we use additional symmetries to extend existing renormalisation inequalities and establish logarithmic fluctuations of the height at a given point. This is consistent with the conjecture that the scaling limit of these height functions is the Gaussian free field.