Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, United Kingdom.
Departamento de Física-CIOyN, Universidad de Murcia, Murcia 30.071, Spain.
Phys Rev Lett. 2013 Sep 6;111(10):100601. doi: 10.1103/PhysRevLett.111.100601. Epub 2013 Sep 4.
Statistical lattice ensembles of loops in three or more dimensions typically have phases in which the longest loops fill a finite fraction of the system. In such phases it is natural to ask about the distribution of loop lengths. We show how to calculate moments of these distributions using CP(n-1) or RP(n-1) and O(n) σ models together with replica techniques. The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter θ fixed by the loop fugacity and by symmetries of the ensemble. We also discuss features of the length distribution for shorter loops, and use numerical simulations to test and illustrate our conclusions.
在三维或更高维度上，统计格点环系通常具有这样的相，其中最长的环填充了系统的有限分数。在这样的相中，很自然地会询问环长度的分布。我们展示了如何使用 CP(n-1) 或 RP(n-1) 和 O(n) σ 模型以及复制技术来计算这些分布的矩。对于宏观环，得到的联合长度分布是泊松-狄利克雷分布，其中参数θ由环逸度和系综的对称性确定。我们还讨论了较短环的长度分布的特征，并使用数值模拟来检验和说明我们的结论。
