论文标题
在没有宏观周期的随机排列中最长循环的精确渐近造物主义
Precise asymptotics of longest cycles in random permutations without macroscopic cycles
论文作者
论文摘要
我们认为，长度为$ n $的ewens随机排列，其周期不超过$ n^β$，$ n^β$，$ 0 <β<1 $，并将渐近行为研究为$ n \ to \ to \ infty $。我们获得有关最长周期长度的联合分布的非常精确的信息；特别是我们证明了一个功能极限定理，其中长周期的累积数量在适当的缩放尺度中收敛到泊松过程。此外，我们证明了关节周期计数与合适的独立泊松随机变量之间的总变异距离的收敛性，最大循环长度明显大于以前所知。最后，我们从中央限制定理中删除了一个多余的假设，该假设是在早期论文中证明的周期总数。
We consider Ewens random permutations of length $n$ conditioned to have no cycle longer than $n^β$ with $0<β<1$ and to study the asymptotic behaviour as $n\to\infty$. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.
