Monotone loop models and rational resonance

Let $T_{n,m}=mathbb Z_ntimesmathbb Z_m$, and define a random mapping $phicolon T_{n,m}to T_{n,m}$ by $phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random walks $phi$ in the limit $m,ntoinfty$, 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 $n^{4/3}$.