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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Durrett R.: Probability: Theory and Examples. The Wadsworth and Brooks/Cole Statistics/Probability Series, Belmont (1991)
Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford Science Publications, Oxford (1979)
Kenyon R., Wilson D.: Resonance in the non-intersecting lattice path model. Probab. Theory Related Fields 130(3), 289–318 (2004)
Hensley D.: The largest digit in the continued fraction expansion of a rational number. Pac. J. Math. 151(2), 237–255 (1991)
Petrov V.V.: Limit Theorems of Probability Theory. Oxford University Press, Oxford (1995)
Richter W.: Local limit theorems for large deviations. Theory Probab. Appl. 2(2), 206–220 (1957)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0285-8