Convergence of the Law of the Environment Seen by the Particle for Directed Polymers in Random Media in the L 2 Region

We consider the model of directed polymers in an i.i.d. Gaussian or bounded environment (Imbrie and Spencer in J. Stat. Phys. 52(3/4), 609-626, 1988; Carmona and Hu in Probab. Theory Relat. Fields 124(3), 431-457, 2002; Comets et al. in Adv. Stud. Pure Math. 39, 115-142, 2004) 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 (Talagrand in Probab. Theory Relat. Fields 110, 177-276, 1998). This improves in some sense a concentration inequality obtained by Carmona and Hu for Gaussian environments. We use this and a local limit theorem (Sinai in Fund. Math. 147, 173-180, 1995; Vargas in Ann. Inst. H. Poincar, Probab. Stat. 42(5), 521-534, 2006) 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.