no. 1
Large deviations for transient random walks in random environment on a Galton-Watson tree
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 159-189.

Nous considérons une marche aléatoire en milieu aléatoire sur un arbre de Galton-Watson. Soit τn le temps d'atteinte du niveau n. Le papier présente un principe de grandes déviations pour τn/n, dans les cas quenched et annealed. Nous étudions ensuite le régime sous-exponentiel, qui fait apparaître un régime polynomial rappelant la dimension 1. Le papier repose principalement sur les estimations de la queue de distribution du premier temps de renouvellement.

Consider a random walk in random environment on a supercritical Galton-Watson tree, and let τn be the hitting time of generation n. The paper presents a large deviation principle for τn/n, both in quenched and annealed cases. Then we investigate the subexponential situation, revealing a polynomial regime similar to the one encountered in one dimension. The paper heavily relies on estimates on the tail distribution of the first regeneration time.

DOI : 10.1214/09-AIHP204
Classification : 60K37, 60J80, 60F15, 60F10
Mots clés : random walk in random environment, law of large numbers, large deviations, Galton-Watson tree 
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     title = {Large deviations for transient random walks in random environment on a {Galton-Watson} tree},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {159--189},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {1},
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     doi = {10.1214/09-AIHP204},
     mrnumber = {2641775},
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Aidékon, Elie. Large deviations for transient random walks in random environment on a Galton-Watson tree. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 159-189. doi : 10.1214/09-AIHP204. http://www.numdam.org/articles/10.1214/09-AIHP204/

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