循环$O(n)$模型在Nienhuis临界点的宏观循环
Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point
摘要：
环$O(n)$模型是一种在六方晶格上随机收集不相交环的模型，它被认为与自旋$O(n)$模型属于同一普适类。Nienhuis曾预言，对于$0\le n\le 2$，环$O(n)$模型在临界参数$x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}$处表现出相变。对于$0<n\le 2$，过渡线已被进一步简化，以将$x<x_c(n)$时具有短环的区域与$x\ge x_c(n)$时具有宏观环的区域分开。本文证明了对于$n\in [1,2]$和$x=x_c(n)$，回路$O(n)$模型存在宏观回路.这是第一个实例中，一个循环$O(n)$模型与$n\neq 1$显示出这样的行为。证明中的一个主要工具是一个新的正关联（FKG）属性，当$n \ge 1$和$0<x\le\frac{1}{\sqrt{n}}$。这个属性意味着，使用最近开发的随机集群模型的技术，以下二分法：要么长循环是指数不可能的，或者原点被任何尺度的循环包围（盒交叉属性）。我们开发了一种“域胶合”技术，使我们能够采用Smirnov的parafermionic可观察到的$x=x_c(n)$和$n\in[1,2]$时，排除第一种选择。
英文摘要：
The loop $O(n)$ model is a model for a random collection of non-intersecting loops on the hexagonal lattice, which is believed to be in the same universality class as the spin $O(n)$ model. It has been predicted by Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model exhibits a phase transition at a critical parameter $x_c(n)=\tfrac{1}{\sqrt{2+\sqrt{2-n}}}$. For $0<n\le 2$, the transition line has been further conjectured to separate a regime with short loops when $x<x_c(n)$ from a regime with macroscopic loops when $x\ge x_c(n)$. In this paper, we prove that for $n\in [1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits macroscopic loops. This is the first instance in which a loop $O(n)$ model with $n\neq 1$ is shown to exhibit such behaviour. A main tool in the proof is a new positive association (FKG) property shown to hold when $n \ge 1$ and $0<x\le\frac{1}{\sqrt{n}}$. This property implies, using techniques recently developed for the random-cluster model, the following dichotomy: either long loops are exponentially unlikely or the origin is surrounded by loops at any scale (box-crossing property). We develop a 'domain gluing' technique which allows us to employ Smirnov's parafermionic observable to rule out the first alternative when $x=x_c(n)$ and $n\in[1,2]$.
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