*Preprint*

**Inserted:** 6 aug 2019

**Last Updated:** 7 aug 2019

**Year:** 2019

**Abstract:**

We investigate the relationship between the $N$-clock model (also known as planar Potts model or $\mathbb{Z}_N$-model) and the $XY$ model (at zero temperature) through a $\Gamma$-convergence analysis as both the number of particles and $N$ diverge. By suitably rescaling the energy of the $N$-clock model, we illustrate how its thermodynamic limit strongly depends on the rate of divergence of~$N$ with respect to the number of particles. The $N$-clock model turns out to be a good approximation of the $XY$ model only for $N$ sufficiently large; in other regimes of $N$, we show with the aid of cartesian currents that its asymptotic behavior can be described by an energy which may concentrate on geometric objects of various dimensions.

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