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

This article is cited in 10 scientific papers (total in 10 papers)

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

Bibliographic databases:

Received: 18.09.2004
Revised: 05.01.2006
Language:

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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\pages 699--711
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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  crossref  mathscinet  zmath  isi  scopus
    2. Dombry C., Guillotin-Plantard N., “A functional approach for random walks in random sceneries”, Electron. J. Probab., 14 (2009), 1495–1512  crossref  mathscinet  zmath  isi  scopus
    3. Pène F., “Transient random walk in $\mathbb Z^2$ with stationary orientations”, ESAIM Probab. Stat., 13 (2009), 417–436  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    5. de Loynes B., “Random walks on directed lattices and Martin boundary”, C. R. Math. Acad. Sci. Paris, 350:1-2 (2012), 87–90  crossref  mathscinet  zmath  isi  scopus
    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  crossref  mathscinet  isi  scopus
    7. Bremont J., “on Planar Random Walks in Environments Invariant By Horizontal Translations”, Markov Process. Relat. Fields, 22:2 (2016), 267–309  mathscinet  zmath  isi
    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  crossref  isi  scopus
    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  crossref  mathscinet  isi  scopus
    10. Ledger S., Toth B., Valko B., “Random Walk on the Randomly-Oriented Manhattan Lattice”, Electron. Commun. Probab., 23 (2018), 43  crossref  zmath  isi  scopus
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