通过XOR技巧研究环O(n)模型中的宏观环
Macroscopic loops in the loop O(n) model via the XOR trick
AI总结：
本文利用XOR技巧、辅助平面图渗流归约及Benjamini--Schramm极限结果，首次在环O(n)模型相图正面积区域证明了宏观环的存在性。
摘要：
环O(n)模型是六角格上不相交环集合上的一族概率测度，由环权重n和边权重x参数化。Nienhuis预测，对于0≤n≤2，该模型表现出两个区域，由x_c(n)=1/√(2+√(2-n))分隔：当x<x_c(n)时，环长度具有指数尾部；而当x≥x_c(n)时，环是宏观的。在本文中，我们证明了关于环O(n)模型中长环存在的三个结果：- 在区域(n,x)∈[1,1+δ)×(1-δ,1]中，其中δ>0很小，从平移不变Gibbs测度采样的构型要么包含一条无限路径，要么有无穷多个环绕每个面的环。在子区域n∈[1,1+δ)和x∈(1-δ,1/√n]中，我们的结果进一步蕴含Russo--Seymour--Welsh理论。这是相图正面积子集中宏观环存在的第一个证明。- 当n=1, x∈(1,√3]时，存在直径与有限域直径相当的环；该区域等价于三角格上Ising模型反铁磁区域的一部分。- 当n∈[1,2], x=1时，环面上存在不可收缩环。证明的主要成分是：(i) 'XOR技巧'：如果ω是短环的集合，Γ是一个长环，那么ω和Γ的对称差必然也包含一个长环；(ii) 将寻找长环的问题归约为证明辅助平面图上的渗流过程没有无限连通分量，该辅助平面图是使用Chayes--Machta和Edwards--Sokal几何展开构建的；(iii) Benjamini--Schramm极限平面图渗流阈值的最新结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
