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 $$\sim n k^{-\beta }$$ , where $$\beta \approx 2$$ , and (ii) the length of the largest cycle is $$\sim n^\alpha $$ , where $$\alpha $$ 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.

中文翻译：

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随机曲折系统中的循环

曲折系统是两个弧系统的并集，表示集合 $$[2n] = \{1, \ldots , 2n\}$$ 在上下半平面中的非交叉配对。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，对于简单生成的曲折系统，大小为 n 的系统中的循环数随 n 线性增长，并且均匀随机曲折系统中最大循环的长度增长为至少 $$c \log n$$ 与 $$c > 0$$ 。我们还提供了数值证据，表明在大小为 n 的简单生成的曲流系统中，（i）长度为 $$k \ll n$$ 的循环数为 $$\sim nk^{-\beta }$$ ，其中 $$\beta \approx 2$$ ，以及 (ii) 最大周期的长度是 $$\sim n^\alpha $$ ，其中 $$\alpha $$ 接近 4/5。