Large deviation inequalities for supermartingales and applications to directed polymers in a random environment

We first show exponential large deviation inequalities for supermartingales, which are optimal in the independent and identically distributed (iid) case. We then apply them to establish new exponential concentration inequalities for the free energy of directed polymers in random environment; as consequences we obtain upper bounds of its rate of convergences (in probability, almost surely, and in L(p)), and give an expression for the free energy in terms of free energies of some multiplicative cascades. To cite this article: Q. Liu, E Watbled C R. A cad Sci. Paris, Ser. I 347 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.