  Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping
$\phi\colon T_{n,m}\to T_{n,m}$ 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 $m,n\to\infty$,
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 $n^{4/3}$.
