Cycles in random meander systems

TL;DR

This paper investigates the cycle structure of random meander systems, revealing linear growth in the number of cycles, logarithmic growth of the largest cycle, and power-law distributions of cycle lengths, with differences across various meander families.

Contribution

It provides the first detailed analysis of cycle counts and lengths in simply-generated and other meander systems, highlighting new growth behaviors and distributions.

Findings

Number of cycles grows linearly with system size n.

Largest cycle length grows at least as c log n.

Cycle length distribution follows a power-law with exponent ~2.

Abstract

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.