Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. Orlando - M. Ruf

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

created by orlando on 06 Aug 2019
modified by cicalese on 17 Sep 2023


Published Paper

Inserted: 6 aug 2019
Last Updated: 17 sep 2023

Journal: Comm. Pure Appl. Math.
Volume: 75
Number: 10
Pages: 2279–2342
Year: 2022

Title changed; formerly 'From the N-clock model to the XY model: emergence of concentration effects'


We investigate the relationship between the $N$-clock model (also known as planar Potts model or $Z_N$-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.