Monotone loop models and rational resonance
| dc.creator | Hammond, Alan | |
| dc.creator | Kenyon, Richard | |
| dc.date | 2008-06-06 | |
| dc.date.accessioned | 2026-07-07T09:43:14Z | |
| dc.date.available | 2026-07-07T09:43:14Z | |
| dc.description | Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping $ϕ\colon T_{n,m}\to T_{n,m}$ 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 $m,n\to\infty$, 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 $n^{4/3}$. | |
| dc.description | 22 pages, 3 figures | |
| dc.identifier | https://arxiv.org/abs/0806.1236 | |
| dc.identifier | http://arxiv.org/abs/0806.1236 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/162480 | |
| dc.subject | Probability | |
| dc.subject | 60Gxx | |
| dc.title | Monotone loop models and rational resonance | |
| dc.type | text |