We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
Publié le : 2010-05-15
Classification:
Random walk in random environment,
Empirical distribution,
Local time,
Self-intersection local time,
60K37,
60J50,
60J55,
60F10
@article{1273584133,
author = {Gantert, Nina and Peres, Yuval and Shi, Zhan},
title = {The infinite valley for a recurrent random walk in random environment},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
volume = {46},
number = {1},
year = {2010},
pages = { 525-536},
language = {en},
url = {http://dml.mathdoc.fr/item/1273584133}
}
Gantert, Nina; Peres, Yuval; Shi, Zhan. The infinite valley for a recurrent random walk in random environment. Ann. Inst. H. Poincaré Probab. Statist., Tome 46 (2010) no. 1, pp. 525-536. http://gdmltest.u-ga.fr/item/1273584133/