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.