Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large deviations for transient random walks in random environment on a Galton–Watson tree

Elie Aidékon

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 1 (2010), 159-189.

Dates
First available in Project Euclid: 1 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1267454113

Digital Object Identifier
doi:10.1214/09-AIHP204

Mathematical Reviews number (MathSciNet)
MR2641775

Zentralblatt MATH identifier
1191.60119

Subjects
Primary: 60K37: Processes in random environments 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F15: Strong theorems 60F10: Large deviations

Keywords
Random walk in random environment Law of large numbers Large deviations Galton–Watson tree

Citation

Aidékon, Elie. Large deviations for transient random walks in random environment on a Galton–Watson tree. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 1, 159--189. doi:10.1214/09-AIHP204. https://projecteuclid.org/euclid.aihp/1267454113


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References

  • [1] E. Aidékon. Transient random walks in random environment on a Galton–Watson tree. Probab. Theory Related Fields 142 (2008) 525–559.
  • [2] K. B. Athreya and P. E. Ney. Branching Processes. Springer, New York, 1972.
  • [3] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25–37.
  • [4] F. Comets and V. Vargas. Majorizing multiplicative cascades for directed polymers in random media. ALEA 2 (2006) 267–277.
  • [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.
  • [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.
  • [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.
  • [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.
  • [10] Q. Liu. On generalized multiplicative cascades. Stochastic Process. Appl. 86 (2000) 263–286.
  • [11] R. Lyons and R. Pemantle. Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20 (1992) 125–136.
  • [12] R. Lyons, R. Pemantle and Y. Peres. Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106 (1996) 249–264.
  • [13] J. Neveu. Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986) 199–207.
  • [14] R. Pemantle and Y. Peres. Critical random walk in random environment on trees. Ann. Probab. 23 (1995) 105–140.
  • [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.)
  • [16] D. Piau. Théorème central limite fonctionnel pour une marche au hasard en environment aléatoire. Ann. Probab. 26 (1998) 1016–1040.
  • [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.