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Monotone loop models and rational resonance

Abstract:

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 ...

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Publisher copy:
10.1007/s00440-010-0285-8
Journal:
Probability Theory and Related Fields More from this journal
Volume:
150
Issue:
3
Pages:
613-633
Publication date:
2011-08-01
DOI:
ISSN:
0178-8051
Language:
English
Keywords:
Pubs id:
pubs:179394
UUID:
uuid:f867dbe0-1fe6-44d9-8277-e3a9cc520bef
Local pid:
pubs:179394
Source identifiers:
179394
Deposit date:
2012-12-19

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