论文标题
$ n $ cycle的随机稳定类型的最小化因素
Random stable type minimal factorizations of the $n$-cycle
论文作者
论文摘要
我们研究了$ n $ cycle的随机最小化因素，即，置换$（1 \，2 \ cdots n）$的因素化，以循环$τ_1，\ ldots，τ_k$，其长度$ \ ell（τ_1），\ eld（τ_1），\ ldots，\ ldots，\ ell（f eld（furs）$ verify filefify the fify thecy_k $ ver（fer（ $ \ sum_ {i = 1}^k（\ ell（τ_i）-1）= n-1 $。通过将单位磁盘上刻有黑色多边形的分解循环相关联，然后又一个又一次地读取周期的循环，我们通过磁盘的有色层压板的过程对最小化分解，这些过程是由红色非交叉的子集制成的，由红色非交叉的和弦划定了黑色或白色。我们的主要结果是，当根据Boltzmann的权重随机选择分解时，该过程的融合是$α$稳定定律的范围内随机选择分解，以（1,2] $吸引$α$稳定的定律。在对此过程的研究中，是最小化因素和大小条件标记的随机树模型之间的两者进行的，其顶点是黑色或白色的。
We investigate random minimal factorizations of the $n$-cycle, that is, factorizations of the permutation $(1 \, 2 \cdots n)$ into a product of cycles $τ_1, \ldots, τ_k$ whose lengths $\ell(τ_1), \ldots, \ell(τ_k)$ verify the minimality condition $\sum_{i=1}^k(\ell(τ_i)-1)=n-1$. By associating to a cycle of the factorization a black polygon inscribed in the unit disk, and reading the cycles one after an other, we code a minimal factorization by a process of colored laminations of the disk, which are compact subsets made of red noncrossing chords delimiting faces that are either black or white. Our main result is the convergence of this process as $n \rightarrow \infty$, when the factorization is randomly chosen according to Boltzmann weights in the domain of attraction of an $α$-stable law, for some $α\in (1,2]$. The new limiting process interpolates between the unit circle and a colored version of Kortchemski's $α$-stable lamination. Our principal tool in the study of this process is a bijection between minimal factorizations and a model of size-conditioned labelled random trees whose vertices are colored black or white.
