Francis Comets, Serguei Popov
Limit law for transition probabilities and moderate deviations
for Sinai's random walk in random environment
(657K, Postscript)
ABSTRACT. We consider a one-dimensional random walk in random environment
in the Sinai's regime. Our main result
is that logarithms of the transition
probabilities, after a suitable rescaling, converge in distribution
as time tends to infinity,
to some functional of the Brownian motion.
We compute the law of this functional when the initial and final
points agree.
Also, among other things, we estimate the probability
of being at time~$t$ at distance at least $z$
from the initial position, when $z$ is larger than $\ln^2 t$, but
still of logarithmic order in time.