Emergence of concentration effects in the variational analysis of the $N$-clock model

TL;DR

This paper analyzes the asymptotic behavior of the $N$-clock model as both the number of particles and $N$ diverge, revealing concentration effects and differences from the $XY$ model through $ abla$-convergence and geometric analysis.

Contribution

It provides a $ abla$-convergence analysis of the $N$-clock model, showing how its continuum limits differ from the $XY$ model and revealing concentration phenomena on geometric objects.

Findings

Energy may concentrate on geometric objects of various dimensions.

The asymptotics feature an energy dominating vortex-vortex interactions.

Different divergence rates of $N$ lead to different continuum limits.

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 of a suitable rescaling of the energy as both the number of particles and $N$ diverge. We prove the existence of rates of divergence of $N$ for which the continuum limits of the two models differ. With the aid of Cartesian currents we show that the asymptotics of the $N$-clock model in this regime features an energy which may concentrate on geometric objects of various dimensions. This energy prevails over the usual vortex-vortex interaction energy.