Monotone loop models and rational resonance

Alan Hammond, Richard Kenyon

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…