论文标题
集中效应的出现在$ n $ -clock模型的变分分析中
Emergence of concentration effects in the variational analysis of the $N$-clock model
论文作者
论文摘要
我们研究了$ n $ -Clock模型（也称为Planar Potts模型或$ \ Mathbb {Z} _N $ -MODEL）和$ XY $模型（在零温度下），通过$γ$ -Convergence分析能量的适当分析，作为粒子数量和$ n $ diverge。我们证明存在两种模型的连续限制的差异率的存在。在笛卡尔电流的帮助下，我们表明，在此制度中，$ n $ Clock模型的渐近学具有一种能量，该能量可能集中于各个维度的几何对象。这种能量占据了通常的涡流涡流相互作用能量。
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.
