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 "-lattice in which the spin field is constrained to take values in a discretization SN of the unit circle S1 consisting of N equispaced points. Our г -convergence analysis consists of two steps: we first fix N and let the lattice spacing " ! 0, obtaining an interface energy in the continuum defined on piecewise constant spin fields with values in SN ; at a second stage, we let N ! C1. The final result of this two-step limit process is an anisotropic total variation of S1-valued vector fields of bounded variation.