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

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