Coarse graining and large-$N$ behavior of the $d$-dimensional $N$-clock model

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