We study the asymptotic behavior of the $N$-clock model, a nearest neighbors ferromagnetic spin model on the $d$-dimensional cubic $\varepsilon$-lattice in which the spin field is constrained to take values in a discretization $\mathcal{S}_N$ of the unit circle $\mathbb{S}^{1}$ consisting of $N$ equispaced points. Our $\Gamma$-convergence analysis consists of two steps: we first fix $N$ and let the lattice spacing $\varepsilon \to 0$, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in $\mathcal{S}_N$; at a second stage, we let $N \to +\infty$. The final result of this two-step limit process is an anisotropic total variation of $\mathbb{S}^1$-valued vector fields of bounded variation.

中文翻译：

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$d$维$N$时钟模型的粗粒度和大$N$行为

我们研究了 $N$-clock 模型的渐近行为，这是 $d$ 维立方 $\varepsilon$-lattice 上的最近邻铁磁自旋模型，其中自旋场被约束为在离散化 $\mathcal 中取值单位圆 $\mathbb{S}^{1}$ 的 {S}_N$ 由 $N$ 个等距点组成。我们的 $\Gamma$ 收敛分析包括两个步骤：我们首先固定 $N$ 并让晶格间距 $\varepsilon \to 0$，获得定义在 $\数学{S}_N$; 在第二阶段，我们让 $N \to +\infty$。这个两步限制过程的最终结果是 $\mathbb{S}^1$ 值的有界变异向量场的各向异性总变异。