Cycles in Random Meander Systems

A meander system is a union of two arc systems that represent non-crossing pairings of the set $\left[2n\right]=\{1,\dots ,2n\}$ 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.