  We consider the model of Directed Polymers in an i.i.d. gaussian or bounded
Environment in the $L^2$ region. We prove the convergence of the law of the
environment seen by the particle. As a main technical step, we establish a
lower tail concentration inequality for the partition function for bounded
environments. Our proof is based on arguments developed by Talagrand in the
context of the Hopfield Model. This improves in some sense a concentration
inequality obtained by Carmona and Hu for gaussian environments. We use this
and a Local Limit Theorem to prove the $L^1$ convergence of the density of the
law of the environment seen by the particle with respect to the product
measure.
