Monotone loop models and rational resonance

Let Tn,m = ℤn × ℤm, and define a random mapping φ: Tn,m → Tn,m by φ(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" φ in the limit m, n → ∞, and show how it depends sensitively on the ratio...