Lipowski Adam, Ferreira António L, Lipowska Dorota
Faculty of Physics, Adam Mickiewicz University in Poznań, 61-614 Poznań, Poland.
Departamento de Física, I3N, Universidade de Aveiro, 3810-193 Aveiro, Portugal.
Entropy (Basel). 2020 Nov 19;22(11):1319. doi: 10.3390/e22111319.
Using simulated annealing, we examine a bipartitioning of small worlds obtained by adding a fraction of randomly chosen links to a one-dimensional chain or a square lattice. Models defined on small worlds typically exhibit a mean-field behavior, regardless of the underlying lattice. Our work demonstrates that the bipartitioning of small worlds does depend on the underlying lattice. Simulations show that for one-dimensional small worlds, optimal partitions are finite size clusters for any fraction of additional links. In the two-dimensional case, we observe two regimes: when the fraction of additional links is sufficiently small, the optimal partitions have a stripe-like shape, which is lost for a larger number of additional links as optimal partitions become disordered. Some arguments, which interpret additional links as thermal excitations and refer to the thermodynamics of Ising models, suggest a qualitative explanation of such a behavior. The histogram of overlaps suggests that a replica symmetry is broken in a one-dimensional small world. In the two-dimensional case, the replica symmetry seems to hold, but with some additional degeneracy of stripe-like partitions.
通过模拟退火算法，我们研究了通过向一维链或方形晶格添加一定比例随机选择的链接而得到的小世界的二分法。在小世界上定义的模型通常表现出平均场行为，而与底层晶格无关。我们的工作表明，小世界的二分法确实取决于底层晶格。模拟表明，对于一维小世界，对于任何比例的额外链接，最优划分都是有限大小的簇。在二维情况下，我们观察到两种情况：当额外链接的比例足够小时，最优划分具有条纹状形状，而当额外链接数量较多时，最优划分变得无序，条纹状形状消失。一些将额外链接解释为热激发并参考伊辛模型热力学的观点，对这种行为给出了定性解释。重叠直方图表明，在一维小世界中复制对称性被打破。在二维情况下，复制对称性似乎成立，但条纹状划分存在一些额外的简并性。
