We study site percolation models on planar lattices including the [m ,4,n ,4] lattice and the square tilings on the Euclidean plane () or the hyperbolic plane (), satisfying certain local constraints on degree‐4 faces. These models are closely related to Ising models and XOR Ising models (product of two i.i.d Ising models) on regular tilings of or . In particular, we obtain a description of the numbers of infinite “+” and “−” clusters of the ferromagnetic Ising model on a vertex‐transitive triangular tiling of for different boundary conditions and coupling constants. Our results show the possibility that such an Ising configuration has infinitely many infinite “+” and “−” clusters, while its random cluster representation has no infinite open clusters. Percolation properties of corresponding XOR Ising models are also discussed.

中文翻译：

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平面晶格上的约束渗流，伊辛模型和异或伊辛模型

我们研究了平面网格上的站点渗流模型，包括[ m，4，n，4]网格和欧几里得平面（）或双曲平面（）上的正方形平铺，满足了度4面上的局部约束。这些模型与Ising模型和Xor Ising模型（两个iid Ising模型的乘积）紧密相关，它们的常规平铺为或。尤其是，我们获得了在顶点传递三角形三角形平铺上铁磁Ising模型的无限“ +”和“-”簇的数量的描述。对于不同的边界条件和耦合常数。我们的结果表明，这样的Ising配置可能具有无限多个无限的“ +”和“-”群集，而其随机群集表示没有无限的开放群集。还讨论了相应的XOR Ising模型的渗透性质。