论文标题
随机曲折系统中的周期
Cycles in random meander systems
论文作者
论文摘要
蜿蜒的系统是两个弧系统的联合，该系统代表上半平面中的$ [2n] = \ {1，\ ldots，2n \} $的非交叉配对。在本文中，我们考虑随机曲折系统。我们表明，对于一类随机曲折系统，对于简单生成的蜿蜒系统， - $ n $的系统中的周期数与$ n $线性增长，并且在均匀随机曲折系统中最大循环的长度至少增长为$ c \ log log n $，带有$ c> 0 $。我们还提供了数值证据表明，在简单生成的蜿蜒曲折系统中，$ n $，（i）长度$ k \ ll n $的循环数为$ \ sim n k^{ - β} $，其中$β\ 2 $，大约2 $，（ii）最大的周期的长度是$ \ sim n^a $ $，$ $ $，$ n $ nis $ nis $ 4/4/4//4//4//4//4//4///4//4//4//4//4//4//4//4/////the。我们将这些结果与其他曲折系统家族的增长率进行了比较，曲折系统的其他家族称为彩虹曲折和类似梳子的曲折，并且表现出明显不同的行为。
A meander system is a union of two arc systems that represent non-crossing pairings of the set $[2n] = \{1, \ldots, 2n\}$ in the upper and lower half-plane. In this paper, we consider random meander systems. We show that for a class of random meander systems, -- for simply-generated meander systems, -- the number of cycles in a system of size $n$ grows linearly with $n$ and that the length of the largest cycle in a uniformly random meander system grows at least as $c \log n$ with $c > 0$. We also present numerical evidence suggesting that in a simply-generated meander system of size $n$, (i) the number of cycles of length $k \ll n$ is $\sim n k^{-β}$, where $β\approx 2$, and (ii) the length of the largest cycle is $\sim n^α$, where $α$ is close to $4/5$. We compare these results with the growth rates in other families of meander systems, which we call rainbow meanders and comb-like meanders, and which show significantly different behavior.
