Monotone loop models and rational resonance

TL;DR

This paper investigates the orbit structure of a random walk on a grid with steps in two directions, revealing how the ratio of grid dimensions influences cycle counts and lengths, especially near rational ratios.

Contribution

It provides a detailed analysis of how the orbit structure of monotone loop models depends on the ratio of grid dimensions, highlighting phase transitions near rational ratios.

Findings

Near rational ratios p/q, about √n cycles of length O(n) are likely.

Far from rationals with small denominators, the number of cycles is bounded.

Typical ratios yield cycles of length approximately n^{4/3}.

Abstract

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 $n^{4/3}$.