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},
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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/

[1] E. Aidékon. Transient random walks in random environment on a Galton-Watson tree. Probab. Theory Related Fields 142 (2008) 525-559. | MR | Zbl

[2] K. B. Athreya and P. E. Ney. Branching Processes. Springer, New York, 1972. | MR | Zbl

[3] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25-37. | MR | Zbl

[4] F. Comets and V. Vargas. Majorizing multiplicative cascades for directed polymers in random media. ALEA 2 (2006) 267-277. | MR | Zbl

[5] A. Dembo, N. Gantert, Y. Peres and O. Zeitouni. Large deviations for random walks on Galton-Watson trees: Averaging and uncertainty. Probab. Theory Related Fields 122 (2002) 241-288. | MR | Zbl

[6] A. Dembo, Y. Peres and O. Zeitouni. Tail estimates for one-dimensional random walk in random environment. Comm. Math. Phys. 181 (1996) 667-683. | MR | Zbl

[7] J. Franchi. Chaos multiplicatif: Un traitement simple et complet de la fonction de partition. In Séminaire de Probabilités, XXIX 194-201. Lecture Notes in Math. 1613. Springer, Berlin, 1995. | Numdam | MR | Zbl

[8] T. Gross. Marche aléatoire en milieu aléatoire sur un arbre. Ph.D. thesis, 2004.

[9] H. Kesten, M. V. Kozlov and F. Spitzer. A limit law for random walk in a random environment. Compos. Math. 30 (1975) 145-168. | Numdam | MR | Zbl

[10] Q. Liu. On generalized multiplicative cascades. Stochastic Process. Appl. 86 (2000) 263-286. | MR | Zbl

[11] R. Lyons and R. Pemantle. Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 (1992) 125-136. | MR | Zbl

[12] R. Lyons, R. Pemantle and Y. Peres. Biased random walks on Galton-Watson trees. Probab. Theory Related Fields 106 (1996) 249-264. | MR | Zbl

[13] J. Neveu. Arbres et processus de Galton-Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 199-207. | Numdam | MR | Zbl

[14] R. Pemantle and Y. Peres. Critical random walk in random environment on trees. Ann. Probab. 23 (1995) 105-140. | MR | Zbl

[15] V. V. Petrov. Sums of Independent Random Variables. Springer, New York, 1975. (Translated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.) | MR | Zbl

[16] D. Piau. Théorème central limite fonctionnel pour une marche au hasard en environment aléatoire. Ann. Probab. 26 (1998) 1016-1040. | MR | Zbl

[17] O. Zeitouni. Random walks in random environment. In Lectures on Probability Theory and Statistics 189-312. Lecture Notes in Math. 1837. Springer, Berlin, 2004. | MR | Zbl

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