Monotone loop models and rational resonance

Abstract

Let $T_{n,m}={\mathbb{Z}}_n\times {\mathbb{Z}}_m$, and define a random mapping $\varphi :T_{n,m}\to T_{n,m}$ by $\varphi (x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random walks'' $\varphi$ in the limit $m,n\to \infty$, and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a rational $p/q$, we show that there are likely to be on the order of $\sqrt{n}$ cycles, each of length O(n), whereas for $m/n$ far from any rational with small denominator, there are a bounded number of cycles, and for typical $m/n$ each cycle has length on the order of $n^{4/3}$.

22 pages, 3 figures