Ising模型中自旋关联在等径图上的普适性
Universality of spin correlations in the Ising model on isoradial graphs
摘要：
我们证明了自旋相关的普遍性的缩放极限的平面伊辛模型的等角图一致有界的角度和Z-不变的重量。具体来说，我们表明，在大规模的缩放限制，即，当网格尺寸$δ$以与逆温度达到临界温度相同的速率趋于零时，在全平面中的两点自旋相关表现为\[ δ^{-\frac{1}{4}}\mathbb{E}\left[σ_{u_{1}}σ_{u_{2}}\right]\ \to\ C_σ^{2}\cdotΞ\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quadδ\to0, \]，其中普适常数$C_σ$和函数$Ξ(|u_{1}-u_{2}|,m)$与晶格无关。质量$m$由关系式$k'-1\sim 4mδ$定义，其中$k'$是Baxter椭圆参数。这些结果，连同为获得它们而发展的技术，足以把前两位作者和C.临界时的洪勒，以及S.C.停在亚临界大质量区域。我们还给出了一个简单的证明，这一事实的无限体积磁化的Z不变模型是独立的网站和晶格。与文献中已有的技术相比，我们简化了对其分歧点附近的离散（大量）全纯旋量的分析，仅依赖于$m=0$的核心$z^{-1/2}$和$m\ne 0$的$z^{-1/2}e^{\pm 2m|z|}$的离散类似物。使推广到等径图，并提供了一个坚实的基础，进一步推广，我们的方法也大大简化了在正方形格设置的证明。
英文摘要：
We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i.e., as the mesh size $δ$ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave as \[ δ^{-\frac{1}{4}}\mathbb{E}\left[σ_{u_{1}}σ_{u_{2}}\right]\ \to\ C_σ^{2}\cdotΞ\left(|u_{1}-u_{2}|,m\right)\quad\text{as}\quadδ\to0, \] where the universal constant $C_σ$ and the function $Ξ(|u_{1}-u_{2}|,m)$ are independent of the lattice. The mass $m$ is defined by the relation $k'-1\sim 4mδ$, where $k'$ is the Baxter elliptic parameter. This includes $m$ of both signs as well as the critical case when $Ξ(r,0)=r^{-1/4}.$ These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S.C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice. As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel $z^{-1/2}$ for $m=0$ and of $z^{-1/2}e^{\pm 2m|z|}$ for $m\ne 0$. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.
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