Abstract
We consider a transient random walk (X n ) in random environment on a Galton–Watson tree. Under fairly general assumptions, we give a sharp and explicit criterion for the asymptotic speed to be positive. As a consequence, situations with zero speed are revealed to occur. In such cases, we prove that X n is of order of magnitude n Λ, with \({\Lambda \in (0,1)}\). We also show that the linearly edge reinforced random walk on a regular tree always has a positive asymptotic speed, which improves a recent result of Collevecchio (Probab Theory Related 136(1):81–101, 2006).
Article PDF
Similar content being viewed by others
References
Collevecchio A. (2006). Limit theorems for reinforced random walks on certain trees. Probab. Theory Related Fields 136(1): 81–101
Coppersmith, D., Diaconis, P.: Random walks with reinforcement. Unpublished manuscript, 1987
Dembo A., Gantert N., Peres Y. and Zeitouni O. (2002). Large deviations for random walks on Galton-Watson trees: averaging and uncertainty. Probab. Theory Related Fields 122(2): 241–288
den Hollander F. (2000). Large deviations, volume 14 of Fields Institute Monographs. American Mathematical Society, Providence
Doyle P.G. and Snell J.L. (1984). Random walks and electric networks, volume 22 of Carus Mathematical Monographs. Mathematical Association of America, Washington
Feller W. (1971). An Introduction to Probability Theory and Its Applications, vol 2, 2nd edn. Wiley, New York
Gross, T.: Marche aléatoire en milieu aléatoire sur un arbre. PhD thesis, 2004
Hu Y. and Shi Z. (2007). Slow movement of random walk in random environment on a regular tree. Ann. Probab. 35(5): 1978–1997
Hu Y. and Shi Z. (2007). A subdiffusive behaviour of recurrent random walk in random environment on a regular tree. Probab. Theory Related Fields 138(3–4): 521–549
Kesten H., Kozlov M.V. and Spitzer F. (1975). A limit law for random walk in a random environment. Compositio Math. 30: 145–168
Lyons R. and Pemantle R. (1992). Random walk in a random environment and first-passage percolation on trees. Ann. Probab. 20(1): 125–136
Lyons R., Pemantle R. and Peres Y. (1995). Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergodic Theory Dyn. Syst. 15(3): 593–619
Lyons R., Pemantle R. and Peres Y. (1996). Biased random walks on Galton–Watson trees. Probab. Theory Related Fields 106(2): 249–264
Menshikov, M., Petritis, D.: On random walks in random environment on trees and their relationship with multiplicative chaos. In: Mathematics and computer science, II (Versailles, 2002), Trends Math., pp. 415–422. Birkhäuser, Basel (2002)
Pemantle R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16(3): 1229–1241
Solomon F. (1975). Random walks in a random environment. Ann. Probab. 3: 1–31
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aidékon, E. Transient random walks in random environment on a Galton–Watson tree. Probab. Theory Relat. Fields 142, 525–559 (2008). https://doi.org/10.1007/s00440-007-0114-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0114-x