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 Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 4, Pages 815–826 (Mi tvp1651)

Short Communications

Transient Random Walks on 2D-Oriented Lattices

N. Guillotin-Plantarda, A. Le Nyb

a Institut Camille Jordan, Université Claude Bernard Lyon 1
b Paris-Sud University 11

Abstract: We study the asymptotic behavior of the simple random walk on oriented versions of $Z^2$. The considered lattices are not directed on the vertical axis but unidirectional on the horizontal one, with random orientations whose distributions are generated by a dynamical system. We find a sufficient condition on the smoothness of the generation for the transience of the simple random walk on almost every such oriented lattices, and as an illustration we provide a wide class of examples of inhomogeneous or correlated distributions of the orientations. For ergodic dynamical systems, we also prove a strong law of large numbers and, in the particular case of independent identically distributed orientations, we solve an open problem and prove a functional limit theorem in the space $\mathscr{D}([0,\infty[,R^2)$ of càdlàg functions, with an unconventional normalization.

Keywords: random walks, random environments, random sceneries, oriented graphs, dynamical systems, recurrence versus transience, limit theorems.

DOI: https://doi.org/10.4213/tvp1651

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English version:
Theory of Probability and its Applications, 2008, 52:4, 699–711

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Revised: 05.01.2006
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Citation: N. Guillotin-Plantard, A. Le Ny, “Transient Random Walks on 2D-Oriented Lattices”, Teor. Veroyatnost. i Primenen., 52:4 (2007), 815–826; Theory Probab. Appl., 52:4 (2008), 699–711

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp1651
• https://doi.org/10.4213/tvp1651
• http://mi.mathnet.ru/eng/tvp/v52/i4/p815

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Guillotin-Plantard N., Le Ny A., “A functional limit theorem for a 2D-random walk with dependent marginals”, Electron. Commun. Probab., 13 (2008), 337–351
2. Dombry C., Guillotin-Plantard N., “A functional approach for random walks in random sceneries”, Electron. J. Probab., 14 (2009), 1495–1512
3. Pène F., “Transient random walk in $\mathbb Z^2$ with stationary orientations”, ESAIM Probab. Stat., 13 (2009), 417–436
4. Guillotin-Plantard N., Prieur C., “Limit theorem for random walk in weakly dependent random scenery”, Ann. Inst. Henri Poincaré Probab. Stat., 46:4 (2010), 1178–1194
5. de Loynes B., “Random walks on directed lattices and Martin boundary”, C. R. Math. Acad. Sci. Paris, 350:1-2 (2012), 87–90
6. Devulder A., Pene F., “Random Walk in Random Environment in a Two-Dimensional Stratified Medium with Orientations”, Electron. J. Probab., 18 (2013), 18, 1–23
7. Bremont J., “on Planar Random Walks in Environments Invariant By Horizontal Translations”, Markov Process. Relat. Fields, 22:2 (2016), 267–309
8. Petritis D., “On the Pertinence to Physics of Random Walks Induced By Random Dynamical Systems: a Survey”, 5Th International Conference on Mathematical Modeling in Physical Sciences (Ic-Msquare 2016), Journal of Physics Conference Series, 738, eds. Vagenas E., Vlachos D., IOP Publishing Ltd, 2016, UNSP 012003
9. Menshikov V M., Petritis D., Wade A.R., “Heavy-Tailed Random Walks on Complexes of Half-Lines”, J. Theor. Probab., 31:3 (2018), 1819–1859
10. Ledger S., Toth B., Valko B., “Random Walk on the Randomly-Oriented Manhattan Lattice”, Electron. Commun. Probab., 23 (2018), 43
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