Monotone loop models and rational resonance

Journal article

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 m/n. For m/n near a rational p/q, we show that there are likely to be on the order of 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 n4/3. © 2010 Springer-Verlag.

Authors: Hammond, A; Kenyon, R

• Journal: Probability Theory and Related Fields
• Volume: 150
• Issue: 3
• Pages: 613-633
• Publication date: 2011-08-01
• DOI: 10.1007/s00440-010-0285-8
• ISSN: 0178-8051
• Language: English
• Keywords: 60D05