Title: Directed Spatial Permutations on Asymmetric Tori Show Chinese title

Title: 定向空间置换在非对称 tori 上

Abstract: We investigate a model of random spatial permutations on two-dimensional tori, and establish that the joint distribution of large cycles is asymptotically given by the Poisson--Dirichlet distribution with parameter one. The asymmetry of the tori we consider leads to a spatial bias in the permutations, and this allows for a simple argument to deduce the existence of mesoscopic cycles. The main challenge is to leverage this mesoscopic structure to establish the existence and distribution of macroscopic cycles. We achieve this by a dynamical resampling argument in conjunction with a method developed by Schramm for the study of random transpositions on the complete graph. Our dynamical analysis implements generic heuristics for the occurrence of the Poisson--Dirichlet distribution in random spatial permutations, and hence may be of more general interest. Show Chinese abstract

Abstract: 我们研究了二维环面上随机空间置换的一个模型，并证明大循环的联合分布渐近服从参数为一的泊松-狄利克雷分布。 我们所考虑的环面的非对称性导致置换中存在空间偏差，这使得我们可以简单地推断出介观尺度循环的存在。 主要挑战在于利用这种介观结构来证明宏观尺度循环的存在及其分布。 我们通过动态重采样论证，结合施拉姆（Schramm）为完全图上的随机换位研究开发的方法实现了这一目标。 我们的动态分析实现了随机空间置换中泊松-狄利克雷分布出现的一般启发式方法，因此可能具有更广泛的兴趣。