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Monotone loop models and rational resonance
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  • Published: 07 May 2010

Monotone loop models and rational resonance

  • Alan Hammond1 nAff2 &
  • Richard Kenyon3 

Probability Theory and Related Fields volume 150, pages 613–633 (2011)Cite this article

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  • 1 Citations

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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 → ∞, 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.

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References

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Author notes
  1. Alan Hammond

    Present address: Department of Statistics, Oxford University, 1 South Parks Road, Oxford, OX1 3TG, UK

Authors and Affiliations

  1. Mathematics Department, New York University, 251 Mercer St., New York, NY, 10012, USA

    Alan Hammond

  2. Mathematics Department, Brown University, 151 Thayer Street, Providence, RI, 02912, USA

    Richard Kenyon

Authors
  1. Alan Hammond
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  2. Richard Kenyon
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Correspondence to Alan Hammond.

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Cite this article

Hammond, A., Kenyon, R. Monotone loop models and rational resonance. Probab. Theory Relat. Fields 150, 613–633 (2011). https://doi.org/10.1007/s00440-010-0285-8

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  • Received: 28 March 2008

  • Revised: 16 March 2009

  • Published: 07 May 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0285-8

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Mathematics Subject Classification (2000)

  • 60D05
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