论文标题
通过XOR Trick循环O（n）模型中的宏观回路
Macroscopic loops in the loop O(n) model via the XOR trick
论文作者
论文摘要
Loop $ o（n）$型号是六角形晶格中非相互循环收集的一系列概率措施，由循环重量$ n $和一个边缘重量$ x $参数化。 Nienhuis预测，该型号以$ 0 \ leq n \ leq 2 $的价格显示了两个由$ x_c（n）= 1/\ sqrt {2 + \ sqrt {2 + \ sqrt {2-n}} $分开的制度：当$ x <x_c（n）$时，loop长度是$ x <x_c（n），$ x _时，$ x c c x c c c c x c c x c c x c c（宏观。 在本文中，我们证明了有关循环$ o（n）$模型中长循环存在的三个结果： - 在[1,1+δ）\ times（1-δ，1] $中，带有$Δ> 0 $小的$（1-δ，1] $，从翻译不变的Gibbs度量中采样的配置将包含无限路径，或者在每个面孔中都包含一个无限的循环。 （1-δ，1/\ sqrt {n}] $我们的结果进一步暗示了russo-seymour-welsh理论。这是在相图的正区域子集中宏观环存在的第一个证明。 - 每当$ n = 1，x \ in（1，\ sqrt {3}] $中，其直径与有限域的循环存在相当；此策略等于三角晶格上Ising模型的一部分。 - $ n \在[1,2]，x = 1 $时，在圆环上存在非合同循环。 证明的主要成分是：（i）“ Xor Trick”：如果$ω$是短路的集合，而$γ$是一个长循环，则对称差异为$ω$，$γ$也必须包括一个长循环； （ii）减少了寻找长循环的问题，以证明使用Chayes和Edwards-Machta和Edwards构建的辅助平面图上的渗透过程 - 界数几何扩展，没有无限连接的组件； （iii）最新的结果是平面图的本杰明·塞拉姆限制的渗透阈值。
The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model: - In the regime $(n,x) \in [1,1+δ) \times (1- δ, 1]$ with $δ>0$ small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime $n \in [1,1+δ)$ and $x \in (1-δ,1/\sqrt{n}]$ our results further imply Russo--Seymour--Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram. - Existence of loops whose diameter is comparable to that of a finite domain whenever $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. - Existence of non-contractible loops on a torus when $n \in [1,2], x=1$. The main ingredients of the proof are: (i) the `XOR trick': if $ω$ is a collection of short loops and $Γ$ is a long loop, then the symmetric difference of $ω$ and $Γ$ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary planar graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm limits of planar graphs.
