Quenched limits for transient, zero speed one-dimensional random walk in random environment
Peterson, Jonathon ; Zeitouni, Ofer
Ann. Probab., Tome 37 (2009) no. 1, p. 143-188 / Harvested from Project Euclid
We consider a nearest-neighbor, one dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that Xn is of order ns for some s<1. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences {nk} and {xk} depending on the environment only, such that Xnk−xk=o(log nk)2 (a localized regime). On the other hand, there exist sequences {tm} and {sm} depending on the environment only, such that logsm/log tm→s<1 and Pω(Xtm/sm≤x)→1/2 for all x>0 and →0 for x≤0 (a spread out regime).
Publié le : 2009-01-15
Classification:  Random walk,  random environment,  60K37,  60F05,  82C41,  82D30
@article{1234881687,
     author = {Peterson, Jonathon and Zeitouni, Ofer},
     title = {Quenched limits for transient, zero speed one-dimensional random walk in random environment},
     journal = {Ann. Probab.},
     volume = {37},
     number = {1},
     year = {2009},
     pages = { 143-188},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1234881687}
}
Peterson, Jonathon; Zeitouni, Ofer. Quenched limits for transient, zero speed one-dimensional random walk in random environment. Ann. Probab., Tome 37 (2009) no. 1, pp.  143-188. http://gdmltest.u-ga.fr/item/1234881687/