$N$ -钟模型变分分析中浓度效应的出现
Emergence of concentration effects in the variational analysis of the $N$-clock model
摘要：
我们研究了$N$ -时钟模型（也称为平面Potts模型或$\mathbb{Z}_N$ -模型）和$XY$模型（在零温度下）之间的关系，通过$Γ$ -收敛分析，当粒子数和$N$都发散时，能量的适当重新标度。我们证明了存在率的发散$N$的连续极限的两个模型不同。借助笛卡尔电流，我们表明，在这个政权的$N$时钟模型的渐近特征的能量，它可能集中在几何物体的各种尺寸。这种能量超过了通常的涡-涡相互作用能。
英文摘要：
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 $Γ$-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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