论文标题
平面图的iSing模型和S层
Ising model and s-embeddings of planar graphs
论文作者
论文摘要
我们讨论了$ \ Mathcal {s} = \ Mathcal {s} _ \ Mathcal {x} $的概念，该图形{x} $带有最近的neighbor ising模型。 $ \ MATHCAL {S} _ \ MATHCAL {X} $的构建基于Kadanoff-Ceva fermions的传播方程的全局复合物值$ \ MATHCAL {x} $的选择。 $ \ MATHCAL {X} $的每一个选择都将所有其他Fermionic可观察物的解释作为$ \ Mathcal {s} _ \ Mathcal {x} $上的所有其他效率可观察到的函数。我们设置了一个通用框架，用于分析S-EmbedDings $ \ Mathcal {s}^δ$，并使用$δ\至0 $。在整个分析过程中，与$ \ Mathcal {s}^δ$相关的功能$ \ MATHCAL {q}^δ$扮演关键角色，这是双分部分二聚体模型术语中所谓的折纸图。特别是，我们对离散表面极限的平均曲率进行解释，$（\ Mathcal {s}^δ; \ Mathcal {q}^δ）$在Minkowski Space $ \ Mathbb r^{2,1} $中作为DIRAC方程中的质量中的质量描述了该模型的连续性极限。 然后，当$ \ mathcal {s}^δ$具有均匀边界的长度/角度和$ \ Mathcal {q}^δ= o（δ）$时，我们将重点关注最简单的情况。作为一种特殊情况，这包括通过其规范的S-件上的双重周期图上的所有关键ISING模型。在此设置中，我们证明了模型的随机群集表示和基本费米子观测值的收敛性的RSW型交叉估计。与已经存在的文献相比，证明依赖于一种新策略，它还提供了对收敛速度的定量估计。
We discuss the notion of s-embeddings $\mathcal{S}=\mathcal{S}_\mathcal{X}$ of planar graphs carrying a nearest-neighbor Ising model. The construction of $\mathcal{S}_\mathcal{X}$ is based upon a choice of a global complex-valued solution $\mathcal{X}$ of the propagation equation for Kadanoff-Ceva fermions. Each choice of $\mathcal{X}$ provides an interpretation of all other fermionic observables as s-holomorphic functions on $\mathcal{S}_\mathcal{X}$. We set up a general framework for the analysis of such functions on s-embeddings $\mathcal{S}^δ$ with $δ\to 0$. Throughout this analysis, a key role is played by the functions $\mathcal{Q}^δ$ associated with $\mathcal{S}^δ$, the so-called origami maps in the bipartite dimer model terminology. In particular, we give an interpretation of the mean curvature of the limit of discrete surfaces $(\mathcal{S}^δ;\mathcal{Q}^δ)$ viewed in the Minkowski space $\mathbb R^{2,1}$ as the mass in the Dirac equation describing the continuous limit of the model. We then focus on the simplest situation when $\mathcal{S}^δ$ have uniformly bounded lengths/angles and $\mathcal{Q}^δ=O(δ)$; as a particular case this includes all critical Ising models on doubly periodic graphs via their canonical s-embeddings. In this setup we prove RSW-type crossing estimates for the random cluster representation of the model and the convergence of basic fermionic observables. The proof relies upon a new strategy as compared to the already existing literature, it also provides a quantitative estimate on the speed of convergence.
