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arXiv · 2005.13365

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

Marco Cicalese·Matthias Ruf·Gianluca Orlando
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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.

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Marco Cicalese, Matthias Ruf, Gianluca Orlando. 2020-05-27. Emergence of concentration effects in the variational analysis of the $N$-clock model. https://arxiv.org/abs/2005.13365

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